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.