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Bring a large pot of water to a boil. Salt the water generously (about 1 tablespoon kosher salt per quart of water), and add the pasta. Cook until al dente according to package directions. Drain and keep warm.
Put the butter and olive oil in a large (12-inch) skillet and place over medium heat. When the butter melts, stir in the shallot and garlic. Cook until aromatic, about 1 minute.
Add the carrots to the pan. Cook 3-4 minutes, until the carrots turn bright orange.
Stir in the broccoli and asparagus. Cover the pan and let them steam 4-5 minutes, until they are bright green. Remove the lid.
Add the peppers to the skillet, stirring and cooking 1-2 minutes. Pour in the cream and season with 1 teaspoon salt. Turn up the heat to medium-high and bring the cream to a boil. Cook the sauce about 5 minutes, until it looks thickened.
Stir the tomatoes, peas, and ½ cup cheese into the sauce and remove from the heat. Add the pasta and gently stir to coat the strands with the sauce.
Transfer to a serving bowl (or serve straight from the pan). Sprinkle the remaining ¼ cup cheese and the basil leaves over the pasta and serve right away.