Cook bacon in a large Dutch oven over medium-high heat until crisp. Remove bacon from pan, reserving 2 teaspoons drippings in pan. Crumble bacon, and set aside.
Add 1/2 cup onions, celery, and squash to drippings in pan; sauté 8 minutes or until vegetables are tender.
Reserve 1 cup corn; set aside. Place the remaining corn and 1 cup milk in a blender; process until smooth.
Add remaining 1 1/4 cups milk, thyme, 1/2 teaspoon salt, and pepper to blender; process just until combined.
Add pureed mixture and reserved 1 cup corn to pan. Reduce heat to medium; cook 5 minutes or until thoroughly heated, stirring constantly.
Stir in 1/8 teaspoon salt. Ladle about 1 1/2 cups soup into each of 4 bowls; top each serving with about 1 tablespoon bacon, 1 tablespoon remaining onions, and 1 tablespoon cheese.