Combine stew meat and water in a large stock pot; bring to a boil. Add tomatoes, diced onion, salt, and pepper. Reduce heat; cover and simmer 2 hours, stirring occasionally. Add cabbage, carrots, celery, rice, and red peppers, if desired; simmer, uncovered, 1 hour, stirring frequently.
Cook ground beef in a skillet over medium heat until browned, stirring to crumble; remove meat with a slotted spoon, reserving drippings in skillet. Sauté chopped onion in drippings until tender; drain well. Combine cooked ground beef and sautéed onion in a small mixing bowl, stirring well; set aside.
Place eggs in a medium mixing bowl; gradually add flour and baking powder, stirring until mixture forms a soft dough. Turn dough out onto a floured surface, and knead 2 minutes.
Divide dough in half. Roll each half to 1/16-inch thickness on a lightly floured surface; cut each into a 16- x 12-inch rectangle. Cut each rectangle into twelve 4-inch squares. Let dough rest on floured surface at room temperature 30 minutes.
Place a heaping tablespoon of ground beef mixture in center of each square. Moisten edges with water; fold squares in half. Press pastry edges firmly together with a fork.
Gently stir filled noodles into soup; cover and simmer an additional 30 minutes. Ladle into individual soup bowls, and serve immediately.