If you can have a cylindrical specimen of your rock you can do either a creep test (constant stress and measure strain over time) or dynamic modulus test (sinusoidal stress and measure sinusoidal strain at a given frequency). In the creep test, you can identify the elastic instantaneous response, then the time-dependent one. After unloading, you can find the recoverable strain (instantaneous and time-dependent). In the dynamic modulus test, if your material is Viscoelastic (VE), then the strain will be lagging the applied stress and you can find the phase angle (time lag/period *360 degrees). With the dynamic modulus amplitude (max stress/max strain) and the phase angle, you can find the elastic (real) and viscous (imaginary) components of your modulus.

The viscosity of igneous rocks is related to their percentage of silica, the more silica the magma will be more viscous. But from there to measure it experimentally, I see it difficult, since you would have to do it with lava, before it solidifies.

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Elastic deformation is by definition completely recoverable. Elastoplastic deformation tends to begin with an elastic phase, this early elastic component is completely recoverable. Reduce the stress and the elastic strain component ‘bouces’ back. A Griggs rig should be able to record that.