Bring a large pot of salted water to a boil, then reduce to a bubbling simmer and cover to keep close to boiling. Heat 3 tablespoons of the oil in a large skillet over medium heat and cook the garlic until fragrant, 1 minute.
Add the carrots and cook, stirring, until they soften slightly, 2-3 minutes. Add the zucchini and asparagus and cook 2 additional minutes. Transfer to a plate and cover with keep warm. Reheat the water, cook the pasta according to the package directions and drain, reserving ¼ cup of the pasta cooking water.
Add the pasta to the vegetables, adding a little of the pasta cooking water to loosen if needed, then add the remaining oil and the cheese. Season with and salt and pepper to taste. Divide among bowls and garnish with basil and additional cheese