Bring a large pot of water to a boil over high heat. Add pasta and 1 Tbsp. salt and cook until tender.
While pasta is cooking, warm oil over medium heat in a large skillet. Add garlic and cook until fragrant, stirring, about 30 seconds. Stir in artichokes, olives, chicken, broth and 1/2 tsp. salt and heat through.
Drain pasta in a colander. Return to pot and stir in chicken mixture and basil. Adjust seasonings, sprinkle with cheese and serve immediately.