To prepare won tons, place first 5 ingredients in a food processor. Process until well-combined, scraping the sides of bowl once. Working with 1 won ton wrapper at a time (cover the remaining wrappers with a damp towel to keep them from drying), spoon about 1 tablespoon turkey mixture into center of each wrapper. Moisten edges of dough with water; bring 2 opposite corners to center, pinching points to seal. Bring the remaining 2 corners to center, pinching points to seal. Pinch 4 edges together to seal. Place the dumplings on a large baking sheet sprinkled with cornstarch (cover with a damp towel to keep them from drying).
To prepare soup, heat oil in a Dutch oven over medium-high heat until hot. Add carrot and 1/4 cup green onions, and sauté for 3 minutes or until tender. Add water and the next 5 ingredients (water through broth), and bring to a boil. Add won tons; reduce heat, and simmer for 6 minutes. Stir in spinach and tomatoes. Serve immediately.