I am a bit confused by your question. Increased sensible heat flux (H) decreases the magnitude of Monin-Obukhov L and thus increases the stability parameter (z/L). Increased H causes instability (it changes the ambient lapse rate), not the other way around.

What do you mean when you say there is a duality in H for stable environments?

Thanks for the quick reply. I understand your question better. You can have many values of z/L for the same H. L is primarily a function of u* (or alternately, wind speed) and H. As the wind changes velocities, L will change. Assume some u* and H on an average summer day. If the wind dies, u* goes down and z/L goes up. H will initially decrease ( H is proportional to u* and L proportional to u* ^3 ). The system will then adjust since E+H+G +outgoing longwave must equal Rnet.

Perhaps it is just wording, but stability is dependent on H, not the other way around.