Bring a large pot of salted water to a boil. Add the broccoli and cook for about 5 minutes until tender. Transfer the broccoli to a bowl and reserve 1 cup of the broccoli cooking water.

In a skillet, heat olive oil over medium heat. Add garlic and red pepper flakes, cooking for 30 seconds until fragrant. Add the cooked broccoli, white beans, and reserved broccoli water. Season with salt and pepper. Cook for about 10 minutes, stirring occasionally, gently mashing some of the white beans to create a creamier texture.

Using the same pot with the broccoli water, add pasta and cook until al dente according to package instructions. Reserve 1 additional cup of pasta water, then drain the pasta thoroughly.

Transfer the drained pasta into the skillet with the broccoli and bean mixture. Add Greek yogurt or cottage cheese and 1 cup Parmesan cheese, stirring everything together until well combined. Add splashes of reserved pasta water if the sauce appears too thick. If using grilled chicken, add it now.

Taste and adjust with additional salt and pepper if needed. Serve immediately, garnished with remaining 1/2 cup Parmesan cheese.