In a large pot of boiling salted water, cook pasta until al dente. Reserve 1/2 cup pasta water; drain pasta and return to pot.
While pasta is cooking, heat oil in a large skillet over medium. Add garlic and red-pepper flakes; cook, stirring occasionally, until garlic is golden, 1 to 2 minutes. Add broccoli and 1/2 cup water; season with salt and pepper. Cover and cook until broccoli begins to soften, about 8 minutes. Uncover and continue to cook until water has evaporated and broccoli is crisp-tender, 1 to 2 minutes more.
To pasta in pot, add broccoli mixture, Parmesan (reserving some for later), and enough pasta water to create a thin sauce that coats pasta. Serve sprinkled with remaining Parmesan.