Cut tofu into squares. Place on a microwave-safe plate and cover with paper towel. Microwave until moisture is evaporated, 3 to 4 minutes.
Heat vegetable oil in a large saucepan over medium-high heat. Cover tofu squares generously with potato starch. Fry in the hot oil until golden brown, 3 to 5 minutes per side. Place in a bowl.
Mix water and 2 teaspoons potato starch together in a bowl to make the thickening agent.
Combine sake, soy sauce, mirin, and fish stock in a saucepan over medium-high heat. Bring to a boil and add carrots and mushrooms. Cook and stir until carrots are tender, about 5 minutes. Stir in the thickening agent. Cook and stir until sauce is thickened and flavors combine, about 2 minutes more.
Pour sauce over the tofu. Top with Welsh onion and ginger.
Substitute chopped green onions for the Welsh onions, if desired.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. Amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.
Per Serving: 268 calories;10.2 g fat;
25.8 g carbohydrates;
13.2 g protein;
0 mg cholesterol;
1708 mg sodium.