Rinse broccoli and trim and discard stem ends. Coarsely chop tops and stems. In a 5- to 6-quart pan over medium heat, stir olive oil and garlic often until garlic sizzles, then stir in broccoli, chili flakes, 1/2 teaspoon salt, 1/4 teaspoon pepper, and 1/2 cup water. Cover and cook, stirring occasionally, until broccoli is very soft and mashes easily with a fork, 20 to 25 minutes. Add 1/4 cup water to pan if broccoli threatens to scorch.
Meanwhile, in a 5- to 6-quart pan over high heat, bring 4 quarts water to a boil; add penne and cook until tender to bite, 10 to 12 minutes. Drain and return to pan.
Stir broccoli mixture into pasta along with cheese. Add more salt and pepper to taste.