Place chicken breasts on cutting board and, using chef’s knife, cut each horizontally into 3 thin cutlets. Place any thicker pieces between two sheets of plastic wrap and pound with a mallet or the bottom of small saucepan. Season chicken on both sides with ¾ tsp of the salt and pepper.
Place flour in shallow bowl and dredge chicken, shaking off excess.
In large skillet, heat 1 tbsp of the butter and 2 tbsp of the oil over medium-high heat. Cook the chicken in batches, flipping as needed, until golden brown. About 4 minutes. Add 1 tbsp butter and the remaining 2 tbsp oil for the second batch. Remove to a plate.
Reduce heat to medium. In same pan, add shallot and cook, stirring, until softened, about 2 minutes. Add garlic and cook until fragrant, about 30 seconds. Stir in chicken broth and simmer until liquid is reduced by half, 3 to 5 minutes.
Reduce heat to low, stir in remaining 3 tbsp butter, lemon juice, capers and salt and cook until thick and velvety. Return the chicken to pan to warm through. Stir in lemon slices and sprinkle with parsley (if using).