Pat chicken breasts dry and season both sides with salt, pepper, and garlic powder.
Place flour in a shallow dish. Dredge each chicken cutlet, shaking off excess.
Heat 2 tablespoons olive oil and 2 tablespoons butter in a large skillet over medium-high heat.
Add chicken in a single layer. Cook 4-5 minutes per side until golden and cooked through (165°F internal temperature). Transfer to a plate.
Reduce heat to medium. Add remaining 1 tablespoon olive oil and minced garlic. Sauté 1 minute until fragrant.
Pour in white wine, scraping up browned bits from pan bottom. Simmer 2 minutes to reduce slightly.
Add chicken broth, lemon juice, and capers. Bring to a gentle boil, then reduce heat to low.
Stir in remaining 2 tablespoons butter, one piece at a time, until melted and sauce is glossy.
Return chicken to skillet, spooning sauce over top. Simmer 2-3 minutes to heat through and let flavors meld.
Garnish with fresh parsley and lemon slices. Serve immediately.