Season chicken with salt and pepper. Mix panko, garlic, lemon zest and parsley in a shallow bowl. Place flour in a separate shallow bowl. Place eggs in a third shallow bowl.
Dip chicken in flour, then in egg, then in panko mixture, shaking off excess. Place chicken on a plate or tray in a single layer.
Melt 1 Tbsp. butter with 2 Tbsp. oil in a large skillet over medium-high heat until foaming. Add half of chicken pieces and cook, turning once, until golden on both sides and cooked through, 5 to 6 minutes total. Transfer to a clean plate or serving platter. Wipe out skillet with paper towel. Repeat with remaining butter, oil and chicken. Serve immediately.