Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs, adding the broccoli, carrots, bell pepper and peas during the last 3 minutes of cooking. Reserve 1 cup cooking water, then drain the pasta and vegetables and return to the pot.
Meanwhile, make the sauce: Heat 2 tablespoons butter in a large skillet over medium-high heat. Add the garlic and cook, stirring, until it begins to soften, about 1 minute. Add the vegetable broth, 1/2 teaspoon salt, and pepper to taste; bring to a simmer and cook until slightly reduced, about 3 minutes. Whisk in the remaining 2 tablespoons butter until melted and the sauce is thickened, about 1 more minute.
Pour the sauce over the pasta and vegetables and add the cheese, parsley and lemon juice. Cook over low heat, stirring, until the cheese melts, about 1 minute, adding enough of the reserved pasta water to loosen the sauce. Season with salt and pepper and top with more cheese.