Soak mushrooms in hot water for 20-30 minutes, until they become softened. Squeeze out excess water. Trim off the stems and thinly slice the caps. You should save the soaking liquid to add to broths or sauces later.
Preheat oven to 375F. Heat oil in an oven-proof, nonstick skillet over medium heat. Add carrots, mushrooms, and onion, saute for about 3-5 minutes, or until carrots have softened. Using hands (or fork), break up the tofu and add to the pan. Using a spatula, spread the vegetables into an even layer on the bottom of the pan. In a small bowl, beat together the eggs with soy sauce and a splash of milk (optional). Pour into the skillet over the tofu and vegetables. Tilt the pan to make sure eggs settles evenly over the vegetable mixture. Cook for 2-3 minutes until the eggs at the edges of the pan begins to set.
Put the pan in the oven and bake for about 8-10 minutes, or until eggs are set. If you want a crispier top, place in the broiler for about a minute or two. Cool before slicing.
While the frittata is in in the oven, make the sauce. In a small pot, combine mirin, soy sauce, and sugar and bring to a boil. Whisk until sugar completely dissolves. In a small bowl, whisk together starch and water until smooth. Pour the slurry into the pan and quickly whisk the sauce. Bring sauce to a boil and remove from heat once the sauce thickens.
Drizzle teriyaki sauce over sliced frittata and serve.