Place the chicken breasts between two sheets of plastic wrap. Use a meat mallet to pound them to about 1/2 inch thick for even cooking.
In a shallow dish, combine flour, garlic powder, salt, and pepper. Dredge each chicken cutlet in the flour mixture, shaking off excess.
Heat olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Cook chicken cutlets in batches for 4-5 minutes per side until golden and cooked through. Remove and set aside.
Lower heat, add remaining butter to the skillet. Once melted, pour in lemon juice, chicken broth, and capers. Stir, scraping up brown bits for flavor.
Reduce heat to low and stir in heavy cream. Mix until sauce is creamy and smooth, simmering for a few minutes to thicken slightly.
Return chicken cutlets to skillet, spoon sauce over them, and simmer an additional 2-3 minutes to absorb flavors.
Transfer chicken to serving platter, drizzle with sauce, garnish with fresh parsley if desired, and serve immediately.