Slice each chicken breast in half lengthwise to get 4 thin cutlets.
Place each cutlet between two sheets of parchment paper or plastic wrap and lightly pound to an even thickness. Season both sides with salt and pepper.
Spread the flour on a plate and dredge the chicken pieces in it, lightly coating both sides. Shake off any excess flour.
In a large skillet over medium heat, heat 3 tablespoons olive oil. Add the chicken pieces (work in batches if necessary) and cook 4-5 minutes on the first side, or until golden brown. Flip and cook another 4-5 minutes, or until cooked through. Transfer the chicken to a plate and cover with aluminum foil to keep warm.
Return the empty skillet to the stove, add 1 tablespoon butter and heat it over medium heat. Add sliced garlic and cook for 20 secs. Add sliced mushrooms and saute, stirring frequently, for about 2 to 3 minutes.
Pour in dry white wine and lemon juice. Cook for 1 minute, stirring constantly, until the sauce reduces slightly. Add thyme leaves, drained capers and stir. Pour in the heavy cream and cook for another minute.
Meanwhile in a small bowl, mix 1 tablespoon of melted butter and 1/2 tablespoon flour. Add this butter mixture to the mushroom mixture and whisk quickly. Bring to a simmer, stirring constantly. Add salt and pepper to taste. When the mixture thickens turn off heat.
Return the cooked chicken cutlets to the skillet, cover with the sauce, and serve over pasta or rice.