Slice chicken breasts in half horizontally, then pound to even thickness. Season with salt and pepper.
Dredge each piece lightly in flour.
Heat 2 tablespoon butter in a cast iron skillet over medium heat. Add chicken and cook 3–4 minutes per side, until golden. Remove to a plate. Most likely will have to cook the chicken in two batches.
Add remaining butter, garlic, and mushrooms to the pan. Sauté until fragrant and softened, about 3 minutes.
Pour in chicken broth and lemon juice. Let simmer 2–3 minutes.
Return chicken to the skillet and spoon sauce over top. Simmer another 3–4 minutes.
Top with capers and serve right from the pan.