Bring a large pot of water to a boil. Season well with salt. Cook the pasta for 2 minutes less than listed on the box, about 8 minutes. Drain well, reserving 1 cup pasta water.
Meanwhile, heat a large skillet over medium-high heat. Add 2 tablespoons olive oil and the garlic and sweat until fragrant, about 1 minute. Add the sausage and cook, breaking it apart with the back of a spoon, until cooked through and beginning to brown, about 8 minutes. Add the peas, asparagus and tomatoes and toss well to coat. Season with 1/4 teaspoon salt. Add 1/2 cup pasta water and simmer for 3 minutes. Add the pasta to the sauce and sprinkle 1 1/2 cups Parmesan on the bare pasta. Toss it with the sauce, adding the remaining 1/2 cup pasta water as needed to create a light tomato sauce. Stir in the thyme. Drizzle with the remaining 2 tablespoons olive oil and 1/2 cup Parmesan cheese.