Make the cutlets: Lay each chicken breast flat and slice it in half horizontally to make 4 thin pieces. Put them between two sheets of plastic wrap and pound to an even 1/4 inch. Pat them dry and season both sides with the salt and pepper.

Dredge in flour: Spread the flour on a plate and press each cutlet into it on both sides, then shake off the excess. A thin, even coat is what browns and later thickens the sauce. A heavy one slides off in the pan.

Sear the chicken: Heat the olive oil and the 1 tablespoon of butter in a large skillet over medium-high heat until it shimmers. Cook the cutlets in two batches, 3 to 4 minutes per side, until deep golden and the chicken reaches 165°F on an instant-read thermometer. Move them to a plate.

Cook the garlic and reduce the wine: Turn the heat to medium and add the sliced garlic to the same skillet. Cook 30 seconds, just until it smells sweet and before it takes on any color. Pour in the white wine and simmer 3 to 4 minutes, scraping the browned bits off the bottom, until the liquid has reduced by about half.

Build the sauce: Add the chicken broth, lemon juice and rinsed capers. Simmer 4 to 5 minutes, until the sauce has reduced by roughly a third and coats the back of a spoon.

Boil the pasta: While the sauce reduces, cook the angel hair in a pot of salted water according to the package directions until al dente, then drain.

Swirl in the cold butter: Take the skillet off the heat and add the cold butter a piece at a time, swirling the pan until each piece melts into the sauce. It should turn glossy and slightly thick. Stir in the parsley.

Warm the chicken and serve: Return the cutlets to the skillet just long enough to warm through and coat them in sauce, about 1 minute. Serve over the drained angel hair with lemon rounds on the side.