Line a sheet pan and set up a breading station: place flour in a shallow dish, beat eggs with milk, salt, and pepper in another dish. The milk is optional but helps create a silkier coating.
Prepare the chicken: if using thick breasts, place them between two sheets of plastic wrap and gently pound to an even 1/4-inch thickness. This ensures quick, even cooking and a tender bite.
Dredge the chicken: coat each piece in flour, tapping off excess. Dip into the egg mixture, letting the excess drip back into the dish.
Cook the chicken: heat half the butter and olive oil in a large skillet over medium-high. When shimmering, add the chicken in a single layer. Cook about 2–3 minutes per side until golden and just cooked through. Remove to a warm plate.
Deglaze and flavor: in the same skillet, add wine (or more lemon juice), scraping up any browned bits. Let it reduce by half, about 1–2 minutes.
Make the sauce: lower heat and whisk in lemon juice, zest, chicken broth, and the remaining butter. Simmer gently to blend the flavors and slightly thicken the sauce.
Finish the dish: return the chicken to the pan to warm through in the sauce for 1–2 minutes. Add capers (if using) and adjust with a pinch more salt or pepper as needed.
Serve: spoon extra sauce over the chicken, garnish with chopped parsley, and serve with a side like a light pasta, sautéed vegetables, or crusty bread to soak up the sauce.