Heat the olive oil in a large pot or Dutch oven over medium heat. Add the diced onion and cook, stirring occasionally, until it turns soft and translucent—about 3 minutes. Stir in the minced garlic and cook for another 30 seconds until fragrant. Be careful not to burn the garlic.
Pour in the vegetable broth and milk, then add the salt, black pepper, and red pepper flakes if using. Bring the mixture to a gentle boil over medium-high heat. Once it’s bubbling, stir in the pasta. Reduce the heat to medium-low so the liquid simmers steadily.
Let the pasta simmer uncovered, stirring occasionally to prevent sticking, for about 6 minutes. Then place the broccoli florets on top of the pasta (don’t stir them in yet—they’ll steam perfectly). Cover the pot and continue cooking for another 4 to 5 minutes, until the pasta is al dente and the broccoli is bright green and tender when pierced with a fork.
Remove the pot from the heat. Add the shredded cheddar, Parmesan, and the butter (if using). Stir vigorously until the cheese is completely melted and the sauce is smooth and creamy. Taste and adjust salt and pepper if needed. If the sauce seems too thick, stir in a splash of milk or broth.
Serve and enjoy