Place the broccoli, pasta, water, yogurt, olive oil, garlic, salt, pepper, red pepper flakes, if using, and water in a large pot or straight-sided skillet and stir to combine. Bring to a boil over high heat. Continue to boil uncovered, stirring occasionally to prevent the pasta from sticking, until the pasta is al dente and almost all the liquid is evaporated, 8 to 9 minutes.
Remove from the heat and stir in the ½ cup cheese. Taste and season with additional salt as needed. Serve in shallow bowls topped with more cheese.
