First energy is not expended, sint energy is always conserved in a process. It is transformed in another form. In grazing, a part of the solar radiation is first transformed in forage, the remaining being transformed into heat. According to FAO (see attached website) "On the basis of these limitations, the theoretical maximum efficiency of solar energy conversion is approximately 11%. In practice, however, the magnitude of photosynthetic efficiency observed in the field, is further decreased by factors such as poor absorption of sunlight due to its reflection, respiration requirements of photosynthesis and the need for optimal solar radiation levels. The net result being an overall photosynthetic efficiency of between 3 and 6% of total solar radiation." The amount of available forage is then between 3% and 6% of the received solar radiation, which can be recorded.

Energy in forage can  be estimated from the growing amount, or the amount eated by the livestock. This amount could be estimated from the number of grazing cattle per day, multiplied by the average amount an animal eats per day. The energy available in a kg of forage can be estimated either from its constituants, or from the heat produced when burning a known amount of  forage, once dried.

I hope these gfew ideas will help.

http://www.fao.org/docrep/w7241e/w7241e05.htm#1.2.1

Perhaps you can find data on how big an area is required to just barely support a cow under the most marginal conditions. Under such conditions all of the free energy of the forage is required to keep the cow alive, and hence all of the free energy acquired by grazing must be expended in new grazing. Under better conditions less area will be required per cow, say a fraction f as much. Then under these better conditions a fraction f of the free energy acquired by grazing must be expended in new grazing. Note that free energy rather than energy is the important quantity: as noted by Claude-Alain Roulet energy is conserved and never changes, and hence can never be expended.

Yes, you can estimate energy expenditure of the animals while doing activity such as when grazin on the pasture, which is by isotope dilution technique - for example using Tritium (TOH) or Deuterium (DOH) or  doubled labelled water. As far as I know and my experience, it is a common and repeatable technique used in wildlife research!!   

There are several papers dealing with this topic, and in my opinion the relationship between heart rate and energy expenditure is the most promising technique in grazing animals.

Some of the literature you could check:

Lachica M, Aguilera JF. Estimation of energy needs in the free-ranging goat with particular reference to the assessment of its energy expenditure by the 13C-bicarbonate method. Small Rumin Res. 2003; 49: 303–318.

Puchala R, Tovar-Luna I, Goetsch AL, Sahlu T, Carstens GE, Freetly HC. The relationship between heart rate and energy expenditure in Alpine, Angora, Boer and Spanish goat wethers consuming different quality diets at level of intake near maintenance or fasting. Small Rumin Res. 2007; 70: 183–193.

Toerien CA, Sahlu T, Wong WW. Energy expenditure of Angora bucks in peak breeding season estimated with the doubly-labeled water technique. J Anim Sci. 1999;77:3096–3105.

Brosh A. Heart rate measurements as an index of energy expenditure and energy balance in ruminants: A review. J Anim Sci. 2007; 85: 1213–1227.

Green J. The heart rate method for estimating metabolic rate: Review and recommendations. Comp Biochem Physiol A Mol Integr Physiol. 2011; 158: 287–304.

Lachica M, Aguilera JF. Methods to estimate the energy expenditure of goats: From the lab to the field. Small Rumin Res. 2008; 79: 179–182.

Butler PJ, Green JA, Boyd IL, Speakman JR. Measuring metabolic rate in the field: the pros and cons of the doubly labelled water and heart rate methods. Funct Ecol. 2004; 18: 168–183.

Kovács L, Jurkovich V, Bakony M, Szenci O, Póti P, Tőzsér J. Welfare implication of measuring heart rate and heart rate variability in dairy cattle: literature review and conclusions for future research. Animal. 2014; 8: 316–330.

Von Borell E, Langbein J, Després G, Hansen S, Leterrier C, Marchant-Forde J, et al. Heart rate variability as a measure of autonomic regulation of cardiac activity for assessing stress and welfare in farm animals—A review. Physiol Behav. 2007; 92: 293–316.

Kumahara H, Schutz Y, Ayabe M, Yoshioka M, Yoshitake Y, Shindo M, et al. The use of uniaxial accelerometry for the assessment of physical-activity-related energy expenditure: a validation study against whole-body indirect calorimetry. Br J Nutr. 2004; 91: 235–243.

Halsey LG, Shepard ELC, Wilson RP. Assessing the development and application of the accelerometry technique for estimating energy expenditure. Comp Biochem Physiol A Mol Integr Physiol. 2011; 158: 305–314.

Halsey LG, Shepard ELC, Quintana F, Laich AG, Green JA, Wilson RP. The relationship between oxygen consumption and body acceleration in a range of species. Comp Biochem Physiol A Mol Integr Physiol. 2009; 152: 197–202.

Halsey LG, Green JA, Wilson RP, Frappell PB. Accelerometry to estimate energy expenditure during activity: best practice with data loggers. Physiol Biochem Zool. 2009; 82: 396–404.

Aharoni Y, Henkin Z, Ezra A, Dolev A, Shabtay A, Orlov A, et al. Grazing behavior and energy costs of activity: A comparison between two types of cattle. J Anim Sci. 2009; 87: 2719–2731.

Brosh A, Henkin Z, Ungar ED, Dolev A, Orlov A, Yehuda Y, et al. Energy cost of cows’ grazing activity: Use of the heart rate method and the Global Positioning System for direct field estimation. J Anim Sci. 2006;84: 1951–1967.

Animut G, Goetsch AL, Aiken GE, Puchala R, Detweiler G, Krehbiel CR, et al. Grazing behavior and energy expenditure by sheep and goats co-grazing grass/forb pastures at three stocking rates. Small Rumin Res. 2005; 59: 191–201.

Halsey LG, Shepard ELC, Hulston CJ, Venables MC, White CR, Jeukendrup AE, et al. Acceleration versus heart rate for estimating energy expenditure and speed during locomotion in animals: Tests with an easy model species, Homo sapiens. Zoology. 2008; 111: 231–241.

You could look also for more recent papers from Arthur Goetsch and his team.

Best regards, and good luck with your search.