In a 6-qt. stockpot, cook linguine according to package directions, adding asparagus and carrots during the last 3-5 minutes of cooking. Drain; return to pot.
Place a large skillet coated with cooking spray over medium heat. Add onion; cook and stir 3 minutes. Add squash, mushrooms and garlic; cook and stir until crisp-tender, 4-5 minutes.
Add cream and broth; bring to a boil, stirring to loosen browned bits from pan. Reduce heat; simmer, uncovered, until sauce is thickened slightly, about 5 minutes. Stir in peas, ham, 2 tablespoons basil and pepper; heat through.
Add to linguine mixture; stir in 1/2 cup cheese. If desired, top with additional basil and cheese.