Bring a large pot of water to a boil.
Meanwhile, add the broccoli florets and carrots to a steamer basket and place it in saucepan filled with the chicken broth or water. Bring the broth to a boil, then reduce the heat to a simmer and cover the pan. Let steam for 5 to 7 minutes, or until the broccoli and carrots are crisp-tender. Remove from heat and set aside until the pasta is cooked.
When the water in the pot is boiling, add the pasta and a heavy pinch of salt (about 1 tbsp) and stir to combine. Cook until al dente, according to package directions. Reserve 1 cup of the pasta water before draining.
In a deep skillet, melt the butter over medium heat. Add the garlic and cook, stirring, until fragrant, about 2 minutes. Add the cooked pasta and steamed vegetables and toss to coat in the butter. Add the shredded cheese, the lemon juice, and ½ cup of the reserved pasta water and gently toss until the cheese melts and is incorporated into the pasta, about 4 minutes to ensure it’s well combined. If your pasta seems a little dry, add ¼ cup of the remaining pasta water at a time and toss into the pasta. Season with kosher salt and black pepper, to taste.
Serve topped with freshly grated parmesan and crushed red pepper flakes, if using.