In a small bowl, whisk together garlic and butter until well combined. Set aside. Place a chicken breast half on a cutting board with the thick side to the right (if you are right-handed) and place your left hand on the chicken breast. Using a very sharp knife, carefully cut the thick side chicken breast about 1/2 inch from the thin side edge, in a horizontal cut. Open the cut chicken breast and spread it out like an open book. Using a meat mallet, gently pound buttered chicken breast until uniform thickness.
Place the flour in a shallow dish, and brush each chicken breast on both sides with flour. Melt the garlic butter in a large skillet over medium heat until it stops foaming, and cook each chicken breast until golden brown on both sides, 6 to 8 minutes per side. Sprinkle each brisket with salt and pepper. Remove the chicken breasts to a bowl and keep warm.
Cook and stir the mushrooms in the same skillet as the chicken until the mushrooms have absorbed the remaining butter in the skillet and have begun to brown around the edges. Add the capers, lemon juice, white wine, and chicken broth, and stir to combine. Reduce to a simmer. Adjust the salt and pepper again, and stir the parsley into the sauce.
Remove chicken breasts to plates and serve sauce over chicken. Garnish each serving with a lemon wedge.