Cook angel hair according to package; reserve 1/2 cup pasta water and drain.
Season chicken with salt and pepper, dredge in flour, shake off excess.
Heat oil in a large skillet over medium-high; cook chicken 3–4 minutes per side until golden and cooked through. Transfer to plate.
Reduce heat to medium; add butter, broth, lemon juice, and capers to skillet. Scrape browned bits and simmer 2–3 minutes.
Return chicken to pan, spoon sauce over, and simmer 1 minute to meld flavors.
Toss pasta with sauce and reserved pasta water as needed; plate pasta and top with chicken. Garnish with parsley and lemon slices.