1. Cut tofu into 3/4-inch-thick slices. Place tofu in a single layer on several layers of paper towels; cover with additional paper towels. Let stand 30 minutes, pressing down occasionally.
2. Combine 1/3 cup water, vinegar, 1 teaspoon sugar, 1/2 teaspoon garlic, 1/2 teaspoon ginger, and crushed red pepper in a small saucepan. Bring to a boil. Add carrot, and remove from heat; let stand 30 minutes. Drain.
3. Remove tofu from paper towels; cut into 3/4-inch cubes. Place tofu in a medium bowl. Combine remaining 1 teaspoon sugar, 1/2 teaspoon garlic, remaining 1/2 teaspoon ginger, soy sauce, and 1 tablespoon oil, stirring with a whisk. Add 1 tablespoon soy sauce mixture to tofu; toss gently. Let stand 15 minutes.
4. Heat a 10-inch cast-iron skillet over high heat for 4 minutes. Add 1 tablespoon sesame oil; swirl to coat. Add rice to pan in a single layer; cook 1 minute (do not stir). Remove from heat; let stand 20 minutes.
5. Heat a large nonstick skillet over medium-high heat. Add 1 teaspoon oil; swirl to coat. Add 1 1/2 teaspoons soy sauce mixture and bean sprouts to pan; sauté 1 minute. Remove sprouts from pan; keep warm. Add 1 teaspoon oil to pan; swirl to coat. Add mushrooms to pan; sauté 2 minutes. Stir in 1 1/2 teaspoons soy sauce mixture; sauté 1 minute. Remove mushrooms from pan; keep warm. Add 2 teaspoons oil to pan; swirl to coat. Add tofu to pan; sauté 7 minutes or until golden brown. Remove tofu from pan; keep warm. Add remaining 1 teaspoon oil to pan; swirl to coat. Add remaining 1 teaspoon garlic and remaining 1 tablespoon soy sauce mixture; sauté 30 seconds. Add spinach to pan; sauté 1 minute or until spinach wilts. Remove spinach from pan; keep warm. Reduce heat to medium. Melt butter in pan. Crack eggs into pan; cook 4 minutes or until whites are set. Remove from heat.
6. Place 3/4 cup rice in each of 4 shallow bowls. Top each serving evenly with carrots, sprouts, mushrooms, tofu, and spinach. Top each serving with 1 egg and 1 teaspoon chili paste. Sprinkle evenly with salt.