Step 1Place the kombu in a pot with the water. To draw out maximum flavor, let it soak for 15 minutes before placing the pot over medium heat -- this will further infuse the water with the flavor-enhancing properties and nutrients of the kelp.
Step 2Remove the pot from the heat as soon as small bubbles begin to break on the surface and at the edge of the pot. Add the katsuobushi, scattering the flakes across the surface of the water. After several minutes, the fish flakes will begin to sink. The larger the flakes, the longer they will take to sink. To keep the stock from tasting fishy, pour it through cheesecloth or a coffee-filter-lined strainer within 3 or 4 minutes of adding the fish flakes. The kombu can be reserved for another use; discard the remaining solids. The stock will keep for up to 3 or 4 days in the refrigerator.
Step 1Place the tofu, miso and egg in a food processor and pulse until creamy and smooth, about 10 seconds. Transfer the tofu mixture to a small bowl. Sprinkle 1 1/2 tablespoons of the cornstarch over the tofu. With a spatula, use cutting and folding motions to mix thoroughly.
Step 2Remove the stems from the mushrooms (save them for enhancing a soup stock) and wipe the caps clean. If the underside of the cap appears to be trapping grit, brush it clean with a cotton-tipped stick (the kind used for cosmetics or medication). Dust the underside of the caps with the remaining 1 tablespoon cornstarch; a pastry brush will simplify the task.
Step 3Stuff the mushroom caps with the tofu mixture, dividing it evenly. Use a butter knife or spatula to press out any air that might be trapped between the mushroom and the filling, and to smooth the surface, slightly mounding the mixture in the center.
Step 4Use a skillet large enough to hold the mushrooms in a single layer. Place the skillet over medium heat and drizzle the oil, swirling it to coat the surface evenly. Place the mushrooms, filling side down, in the skillet. Press down on the mushrooms ever so slightly with a broad, flat spatula. Hold for a few seconds to ensure the filling adheres to the mushroom caps, then sear the mushrooms for 1 minute undisturbed, or until the filling is very lightly crusted over.
Step 5Flip the mushrooms so that the filling faces up. Again, press lightly on the mushrooms and hold for a few seconds. Lower the heat slightly and add the mirin, soy sauce and stock. When the liquid begins to bubble, flip the mushrooms again, so the filling is face down. Raise the heat to medium-high and simmer for a few minutes until the skillet juices have reduced and thickened, about 3 to 4 minutes.
Step 6Remove the mushrooms from the skillet and arrange them on individual plates or on a single platter. Turn some mushrooms so that the light filling is visible, and others so that the dark caps show. Pour any skillet juices over the mushrooms. Serve warm or at room temperature sprinkled with the sea salt and sansho pepper.