Melt butter in a Dutch oven over medium heat. Add onion, and sauté 5 minutes or until tender. Add thyme; sauté 1 minute or until fragrant.
Stir in broth, pumpkin, and squash. Increase heat to medium-high; bring to a boil. Reduce heat to low, and simmer 6 minutes. Remove from heat, and let cool 10 minutes.
Process soup, in batches, in a food processor or blender until smooth. Return soup to Dutch oven. Stir in half-and-half and remaining ingredients. Cook, over low heat, stirring often, 3 minutes or until thoroughly heated. Serve immediately.
Note: We tested with Publix Cooked Squash. Be sure to use unsweetened squash puree for best results.