Cook spaghetti in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, season chicken with its salt and pepper. Heat 1 tablespoon oil in a large skillet and cook chicken for 5 to 7 minutes until browned and at least 165°F; transfer to a plate.
Lower heat and add remaining oil, garlic, anchovies and crushed red pepper. Stir for 1 minute until the anchovies start dissolving, then add tomatoes, olives, capers and broth.
Simmer for 8 to 10 minutes until slightly thickened. Return chicken and add spaghetti with 1/4 cup reserved pasta water.
Toss over low heat for 1 to 2 minutes until the pasta is tender and the sauce coats the strands, adding more water if needed, then serve.