Slice each chicken breast horizontally to make two thinner cutlets (or butterfly and pound to about 1/2-inch thick). Pat dry with paper towels. Season both sides with kosher salt and black pepper.

Dredge

Place the flour in a shallow dish. Beat the eggs in a second shallow bowl. Lightly coat each cutlet in flour, shake off excess, dip into beaten egg, then shake off excess again (you may skip the second flour for a lighter crust).

Sear

Heat a large skillet over medium-high heat. Add 2 tablespoons olive oil and 2 tablespoons butter. When butter foams and oil shimmers, add chicken cutlets in a single layer (work in batches if needed). Sear 3-4 minutes per side until golden brown and cooked through (internal temperature 165°F / 74°C). Transfer cooked cutlets to a warm plate and tent with foil.

Cook pasta

While chicken cooks, bring a large pot of water to a rolling boil. Add 1 tablespoon salt and the angel hair. Cook according to package directions (usually 3-4 minutes) until al dente. Reserve 1/2 cup pasta cooking water, then drain and keep warm.

Make sauce

Reduce heat under the skillet to medium. If pan is very brown, add 1–2 tablespoons of the white wine or chicken broth and scrape up browned bits with a wooden spoon to deglaze. Add the minced garlic and sauté 30 seconds until fragrant (do not let it brown).

Pour in the remaining white wine (or chicken broth) and bring to a simmer. Cook 2–3 minutes to reduce slightly, scraping up any stuck bits. Stir in the lemon juice and capers, then simmer 1 minute.

Turn the heat to low. Whisk in 2 tablespoons cold butter, one tablespoon at a time, until the sauce emulsifies and becomes glossy. If the sauce seems too tight, add 1–2 tablespoons of reserved pasta water to loosen. Taste and adjust seasoning with a pinch of salt and more pepper if desired.

Finish and serve

Return the chicken to the skillet and spoon sauce over each piece to rewarm for about 1 minute. Stir in chopped parsley into the sauce.

Toss the drained angel hair with a tablespoon of sauce and, if needed, additional reserved pasta water to coat. Divide pasta among plates, place a chicken cutlet on top, spoon extra lemon-caper sauce over chicken and pasta.

Garnish with lemon slices and extra parsley. Serve immediately.