Heat oil in a large nonstick skillet over medium-high heat. Add the mushrooms, carrot, onions, ginger, and garlic; sauté 2 minutes. Stir in broth, soy sauce, sake, sugar, and sambal oelek. Cover, reduce heat, and simmer for 10 minutes, stirring occasionally.
Combine the cornstarch, water, and vinegar, stirring well with a whisk. Add cornstarch mixture to broth mixture, stirring well; bring to a boil. Cook 1 minute or until thick, stirring constantly. Remove from heat; add tofu. Cover and let stand 10 minutes. Serve over rice.