Steam potato, covered, for 20 minutes or until tender. Drain and return to pan; partially mash with a potato masher.
Preheat oven to 375°.
Steam the broccoli, sliced carrot, and chopped onion, covered, for 10 minutes or until tender.
Combine the potato, steamed vegetables, milk, 1/2 cup cheese, basil, salt, and pepper in a large bowl. Spoon the potato mixture into a 2-quart-deep baking dish coated with cooking spray, and top with 1/4 cup cheese. Bake potato mixture, uncovered, at 375° for 20 minutes.