In a large saucepan, cook linguine according to package directions, adding the broccoli, cauliflower and carrot during the last 5 minutes of cooking.
Meanwhile, in another saucepan, saute mushrooms in butter until tender. Stir in the flour, thyme, salt and pepper until blended; gradually add milk. Bring to a boil; cook and stir for 2 minutes or until thickened. Stir in cheese.
Drain linguine and vegetables; transfer to a serving bowl. Add ham and sauce mixture; toss to coat.