Season chicken breasts with salt and pepper. Place flour on a plate. Beat eggs in a shallow baking dish. Place panko breadcrumbs and parsley in another shallow dish.
Working with 1 chicken breast at a time, dredge in flour, shaking off excess. Dip into egg mixture, turning coat evenly and coat with panko, pressing gently. Transfer to a baking sheet.
Add oil and butter in a large skillet over medium-high hear. Add chicken breast to skillet and cook until golden brown on both sides, about 8-10 minutes. Transfer to a paper towel-lined plate and set aside.
Heat butter until foaming and starting to brown, about 3 minutes. Add the rest of the ingredients.
Drizzle the sauce over chicken and serve with lemon wedges.
