Prepare the chicken thighs by trimming off any excess fat. Lay them on a flat surface and cover with a piece of plastic wrap. Use a mallet to pound the chicken to approximately ½ inch thickness.
Place the finely grated Parmesan cheese, salt, and pepper on a plate and toss together to combine. Press each chicken thigh into the cheese mixture coating it in cheese. Set aside.
Place 1 tablespoon of butter and the olive oil in a large skillet and heat over medium high heat. Once hot add the chicken thighs and cook for approximately 3 minutes on each side, until the outside of golden brown. Set cooked chicken aside.
Add the shallots, garlic, capers, and the remaining butter to the skillet and cook for approximately 3 minutes.
Add the white wine and lemon juice and continue to cook, scraping up any browned bits from the bottom of the pan. Cook for 5 minutes before returning the chicken to the skillet and continuing to cook for 5 - 7 minutes, or until the internal temperature of the chicken is 165 degrees F.