Cook the pasta according to package directions. During the last 3 minutes of cooking, add the broccoli florets to the same pot.
Reserve some of the pasta water before draining, then drain the pasta and broccoli. (If you're serving little ones who prefer things plain, set aside some pasta and broccoli here.)
In the same pot, heat the olive oil over medium heat. Add the minced garlic and saute for about 30 seconds, until fragrant.
Add the onion powder, garlic powder, mustard powder, salt, and broth. Simmer for 3–5 minutes.
Stir in the half and half, parmesan, cheddar, and nutritional yeast. Bring to a low simmer for 3–5 minutes, until the sauce is smooth and slightly thickened.
Fold the pasta, broccoli, and white beans into the sauce. Toss to coat, adding a splash of reserved pasta water if needed to loosen things up. Taste and add more salt if needed, then serve warm.