Bring a large pot of water with the tablespoon salt to a boil. Season chicken with its salt and pepper, dredge lightly in flour and shake off excess.
Heat oil in a large skillet over medium heat and cook cutlets for 4 to 5 minutes per side until golden and at least 165°F. Transfer to a board and rest.
Add wine and broth to the skillet and scrape up browned bits. Simmer for 3 minutes, then add lemon juice and capers and cook for 1 minute.
Meanwhile, cook angel hair according to its package time. Reserve 1/2 cup pasta water and drain; remove sauce from heat and swirl in cold butter.
Toss pasta with the sauce, adding reserved water a tablespoon at a time if needed. Slice chicken and serve over the lemon-caper noodles.