Melt the butter in a Dutch oven or similar large, sturdy pot over medium heat. Once the butter is melted, add the onion, salt, and black pepper. Saute until the onion begins to soften, about 4 minutes. Do not let the onion brown (adjust the heat as needed if it starts to brown).

Add the garlic, Italian seasoning, and red pepper flakes. Stir and cook just until fragrant, about 30 seconds.

Slowly stir in the broth and milk. Bring to a boil, then add the pasta. Let boil for 5 minutes, stirring periodically to keep the pasta from sticking to the pot, then stir in the broccoli florets.

Continue boiling until the pasta is al dente, about 5 to 7 minutes more. Stir often to make sure the pasta cooks evenly. If at any point the pasta looks too dry, splash in additional broth as needed. The pasta will look a little liquidy at first but will thicken up as it sits.

Stir in the lemon juice, Parmesan, and parsley. Taste and adjust seasoning as you like. Let rest 5 minutes, then stir once more. Enjoy hot, with a sprinkle of extra Parmesan and parsley as desired.