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Heat a large pot over medium heat. Once the pot is hot, add butter and swirl to coat the bottom.
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Add carrots and onions to the pot; cook, stirring frequently, until slightly softened, 4-5 minutes.
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Add the cauliflower, broccoli, and zucchini to the pot; cook, stirring frequently, 3-4 minutes.
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Break spaghetti into quarters. Add the spaghetti, peas, broth, and salt to the pot. The broth should just cover the spaghetti and vegetables.
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Bring to a boil over high heat; reduce heat to a simmer, and cook, stirring occasionally, until pasta is cooked through and most of the broth is absorbed, 8-10 minutes.
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Meanwhile, finely grate the Parmesan.
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Trim off and discard the ends of the green onions; cut crosswise into ¼-inch pieces.
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Once the pasta is done, remove from heat. Stir in most of the green onions and the Parmesan (save some of each for garnish) until the cheese is melted.
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To serve, divide the pasta between bowls. Top with reserved green onions and Parmesan.