Slice each chicken breast horizontally into 2 thin cutlets (or buy thin-sliced cutlets). Place between plastic wrap and pound gently to an even 1/2-inch thickness. Season with 1/2 teaspoon salt and the pepper. Even thickness is what keeps this fast — thin cutlets cook through in minutes.

Spread the flour in a shallow dish. Dredge each cutlet lightly, shaking off excess. Heat 2 tablespoons olive oil in the large skillet over medium-high heat until shimmering. Sear the cutlets in two batches, 2–3 minutes per side, until deeply golden. Transfer to a plate — they do not need to be fully cooked through yet.

Bring the large pot of salted water to a boil and cook the angel hair 1 minute shy of the package directions. Reserve 1/2 cup pasta water, then drain. Angel hair cooks in 4–5 minutes, so this runs in parallel with the sauce.

Reduce the skillet heat to medium. Add the remaining 1 tablespoon olive oil, then the shallot; cook 2 minutes until softened. Add the garlic; cook 30 seconds. Pour in the white wine, scraping up every browned bit — those bits are the flavor. Add the chicken broth, lemon juice, and capers. Simmer 4–5 minutes until reduced by about a third.

Take the skillet OFF the heat. Add the cold butter a piece at a time, swirling until each melts before adding the next — this is what makes the sauce glossy instead of oily. Stir in the parsley. Return the cutlets and any juices to the pan, turning to coat; simmer gently 1–2 minutes until the chicken is cooked through.

Add the drained angel hair to the skillet, tossing with tongs and splashes of reserved pasta water until every strand is coated in sauce. Serve immediately with lemon slices and extra parsley.