Discard mushroom stems and cut the caps into ½-inch pieces. Spread the mushroom pieces in a 6-quart (or larger) slow cooker. Add carrots, bamboo shoots and tofu to slow cooker; sprinkle with white pepper. Top with cabbage.
Combine 4 cups water, broth, white (or rice) vinegar, red-wine vinegar, soy sauce, chile-garlic sauce and ginger in a bowl; add to the slow cooker.
Cover and cook for 4 hours on High or 7 to 8 hours on Low.
Whisk the remaining ⅓ cup water, cornstarch and sesame oil in a bowl. Stir into the soup. If using the Low setting, turn to High. Cover and cook for 20 minutes. Stir in noodles, cover and cook for 10 minutes more. Serve topped with scallions and with more soy sauce and chile-garlic sauce, if desired.