Combine broth, 2 cups water, the ginger, garlic, and 3 tbsp. soy sauce in a medium saucepan. Bring to a simmer over medium heat; cook 10 minutes.
Mix the remaining 1 tbsp. soy sauce and 1 tbsp. chili sauce in a small bowl; set mixture aside.
Cut tofu crosswise into 8 slabs, each 1/2 in. thick. Heat oil in a large nonstick frying pan over medium heat. Cook tofu, turning once, until lightly browned, about 8 minutes. Remove from heat; pour soy-chili mixture over tofu and turn gently to coat.
Add noodles and carrots to broth and cook, stirring occasionally, until just tender, 3 to 4 minutes. Stir in spinach and cook just until wilted. Ladle noodles, broth, and vegetables into bowls and top each with 2 slabs of tofu. Drizzle tofu with more chili sauce if you like.
*Available refrigerated at well-stocked grocery stores and Asian markets.