1/2 cup of freshly grated Parmigiano-Reggiano cheese
Heat 1/4 cup oil in a large pot over medium-high heat until hot. Add garlic and cook, stirring, until barely golden, about 30 seconds. Add carrots, reduce heat to medium, and cook, stirring occasionally, until beginning to soften, 3 to 4 minutes. Add mushrooms and cook, stirring occasionally, until beginning to soften, 2 to 3 minutes. Add sliced asparagus stalks and cook for 2 minutes, then add the asparagus tips, peas, beans and scallions. Season with sea salt and pepper. Cook, stirring, until all the vegetables are just tender, about 2 minutes more. Remove from the heat.
Meanwhile, bring 6 quarts of water to a boil in another large pot. Add kosher salt and pasta; cook until just tender, 10 to 12 minutes (or according to package directions). Drain, reserving 1/2 cup of the cooking water.
Add the pasta and 1/4 cup of the reserved cooking water to the vegetables, stirring and tossing over medium heat to mix well. Cover, reduce heat to low, and allow to steam together for 2 minutes. Stir in the remaining 2 tablespoons olive oil and a splash or two more of the reserved cooking water if necessary to loosen the sauce. Stir in the parsley, basil and mint. Serve with Parmigiano.