Preheat a 16-inch cast-iron skillet over medium-high heat (or the coals of a camp fire). Season inside cavity and exterior of trout with salt and pepper. Place lemon rings, thyme sprigs (reserving a few), and parsley sprigs inside trout cavities.
Heat oil in pan and cook country ham until browned and fat is rendered. Remove ham and set aside for another use, reserving fat in pan. Place trout in pan and cook until bottom side of fish begins to turn golden. Using a large spoon or spatula, carefully flip fish to brown opposite side. As it begins to brown, drain off oil and add butter, garlic, and remaining thyme sprigs to pan. As butter begins to foam, tilt pan slightly and use a large spoon to baste butter over trout. (This will help cook the trout fully.) Check if trout is cooked through by inserting a knife into it—if the knife comes out clean and is warm, the trout is finished. Carefully remove from pan and serve immediately.