Since the Compton scattering process is where a high energy photon scatters off an electron, would it be a plausible result for the energies of the scattered photon and the scattered electron to be entangled?
The photon and electron will definitely be entangled after scattering.
This is easy to see conceptually in two ways. First way: we can view this process as measuring the photon's position, with the electron playing the role of the apparatus. As usual, after the measurement, the photon and the apparatus will be entangled.
Second way: consider running a million experiments where one billiard ball scatters off another. The final momenta and energies of the balls will be distributed differently depending on the exact place where the balls hit. So there will be a classical correlation between the two balls.
Now replace this classical ensemble with a quantum superposition, and the classical correlation becomes the entanglement. What do I mean by that? Take those million initial states from the previous paragraph and consider a quantum superposition of all of them. That's the analog of the initial state in Compton scattering, since both the photon and electron are in a superposition of different positions. Now, of course time evolution in quantum mechanics is linear, so the final state of the billiard balls will be a superposition of all the million final states, i.e. an entangled state. (Of course you don't really need a million to get entanglement, even just two would suffice.)