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Place the mushrooms in a medium skillet. Add enough water to cover; if they float up, swish them around to make sure they’re wet all over. Boil for 10-15 minutes, or until most (if not all) of the stems are tender. Trim any tough stems off, then halve the mushrooms.
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In the same skillet, heat some oil over medium high heat. Add the mushrooms and saute for 5-10 minutes, or until golden and fragrant. Add the soy sauce, tossing the mushrooms, and cook until darkened. Pull off the heat and mix in the sesame oil.
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In another skillet, heat some oil over low heat. Add the egg, sprinkle with salt, and cook until the top is set. Cool, then slice into 4 strips.
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Place a bamboo sushi rolling mat on a cutting board, and drape a piece of plastic wrap big enough to cover the mat on top. Fill a small bowl with water and keep handy.
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Place a sheet of seaweed, shiny side down, on the plastic wrap. Add about ⅓ cup rice, keeping the top and bottom edges free of rice, and use your fingers to break up and spread the rice. Don’t worry if it looks a little patchy!
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Place two rows of mushrooms in the middle of the rice, stretching from the left to the right edge. Press them into the rice gently to help them stick. Drape a strip of egg over the mushrooms. Place a few avocado slices overlapping the egg, closer to the bottom half of your sushi canvas than the top half. Add 2 pieces of cucumber right above the avocado, and 1 green onion above that. Drape pieces of pickled ginger over the avocado.
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Starting from the bottom end, roll up the sushi tightly using the bamboo mat. You can use one hand to hold in the cucumber/green onion while you roll if the other if you need to. Once you get a few inches away from the top edge, start to keep the bamboo mat out of the rolling process, and use the plastic wrap to guide the rest. Once the sushi is sealed, give it a few squeezes to make sure it’s sticking together, then unwrap from the plastic wrap and sushi mat, and let cool seam side down.
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Using a serrated knife, slice the sushi. I’d recommend thicker slices, since the sushi stays together better that way. Serve at room temperature. I wouldn’t recommend refrigerating, since the rice will get hard.