Place the tofu on several layers of paper towels, and cover with additional paper towels. Let stand 30 minutes, pressing down occasionally. Place tofu in a large bowl, and mash with a fork until smooth. Set tofu aside.
Combine boiling water and mushrooms in a bowl; cover and let stand 20 minutes or until soft. Drain. Place the mushrooms and water chestnuts in a food processor; pulse 5 times or until minced. Add mushroom mixture, carrot, and the next 7 ingredients (carrot through 4 teaspoons cornstarch) to tofu; stir well.
Working with 1 won ton wrapper at a time (cover remaining wrappers to keep them from drying), spoon 1 teaspoon tofu mixture into center of each wrapper. Moisten edges of wrapper with water; bring 2 opposite corners to center, pinching points to seal. Bring remaining 2 corners to center, pinching edges together to seal. Place dumplings, seam sides up, on a large baking sheet sprinkled with 1 teaspoon cornstarch (cover loosely with a towel to keep them from drying).
Arrange one-third of the dumplings in a single layer in a vegetable steamer coated with cooking spray. Steam dumplings, covered, for 15 minutes. Remove the dumplings from steamer; set aside, and keep warm. Repeat the procedure with the remaining dumplings.
Combine 1/2 cup soy sauce and 1/4 cup water in a small bowl. Serve with dumplings.