Heat 1/4 cup oil in large saucepan over medium-high heat. Add onion and carrot; cook 2 to 3 minutes or until onion is tender. Add garlic; cook 1 minute more or until fragrant.
Add water, tomato sauce, tomato paste and pepper sauce to saucepan; stir to combine. Reduce heat and simmer 15 minutes or until thickened slightly, stirring occasionally.
Remove soup from heat. Carefully blend mixture in saucepan with hand-held immersion blender until smooth. Stir in milk and keep warm; set aside.
Make grilled cheese dumplings by mixing mozzarella and American cheeses in medium bowl and mashing together with a fork. Fill each dumpling wrapper with 1 to 1-1/2 teaspoons cheese mixture in the center and run a wet finger around the edge of the dough. Fold in half and press edges to seal. Using a fork, press firmly around edges on both sides to completely seal dumpling.
Cook dumplings in batches. Place 1 to 2 tablespoons oil in large nonstick saute pan and begin frying dumplings. Once browned on first side, turn over and add 2 tablespoons water to pan and cover immediately. Allow steam to finish cooking, about 2 minutes. Remove cover and allow water to evaporate and cook 1 minute more until golden brown. Continue cooking remaining dumplings.
Place dumplings in soup and sprinkle with Parmesan cheese. Dumplings also may be served on the side, if desired.