Bring a large pot of water to boil over high heat. Warm oil over medium heat in a large skillet. Add garlic and cook, stirring, until fragrant, about 30 seconds. Stir in bread crumbs, crushed red pepper and 1/2 tsp. salt and cook, stirring, until bread crumbs are toasted, 2 to 3 minutes. Remove from heat.
Add 1 Tbsp. salt and pasta to pot with boiling water and cook until almost tender. Add broccoli to pot and continue to boil until broccoli and pasta are both tender, about 2 minutes longer. Drain and return pasta and broccoli to pot.
Add chicken broth and chickpeas to pot with pasta and broccoli, return pot to heat and stir until broth is heated through, about 2 minutes. Add bread-crumb mixture to pot with pasta mixture and toss to coat. Serve immediately.