Bring a large pot of salted water to a rolling boil.
Add chickpea pasta to the water and cook for 2 minutes less than the package directions indicate for al dente.
Add the broccoli florets directly into the boiling pasta water for the final 3 minutes of the pasta's cooking time.
Reserve 1/2 cup of the starchy pasta water, then drain the pasta and broccoli.
In the same pot over low heat, sauté the minced garlic in olive oil for 1 minute until fragrant.
Return the pasta and broccoli to the pot.
Whisk the Greek yogurt and Parmesan cheese together in a small bowl, then fold into the pasta mixture.
Gradually add the reserved pasta water while stirring until the sauce reaches a creamy consistency.
Season with salt, black pepper, and red pepper flakes.
Remove from heat immediately to prevent the yogurt from curdling and serve.