How to Make It
Preheat the oven to 375º. In a bowl, toss the mushrooms and soy sauce. Let stand for 5 minutes; drain.
In a 9-inch ovenproof nonstick skillet, heat the oils. Press the rice into the skillet, about 1/4 inch thick. Cook over moderately high heat until the bottom is golden, 10 minutes. Top with the tofu in a single layer and drizzle with unagi sauce. Top with the shiitake, edamame, cheese and bean sprouts and season with salt. Sprinkle with togarashi and sesame seeds.
Transfer the skillet to the oven and bake on the top shelf for 15 minutes, until the top is golden. Slide the pizza onto a platter; top with the bonito, shiso and cilantro and serve.
Sweet-salty unagi sauce is available at Japanese markets.