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Bring a large pot of salted water to a boil.
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Put the olive oil and the butter in a large skillet over medium-high heat until butter is melted. Add snap peas, asparagus, English peas, cut tomatoes and onion. Cook until vegetables are barely tender (but not too soft or mushy), 3 to 4 minutes. Stir in garlic and shrimp and cook until shrimp is pink. Season with salt and pepper; set aside.
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Drop pasta into boiling water and cook until al dente about 8 minutes. Drain well and transfer pasta to a large bowl. Immediately toss pasta with vegetables, Parmigiano-Reggiano, heavy cream and herbs. Season generously with salt and pepper, if needed.