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Make the steamed buns: In the bowl of a stand mixer, combine the water and active dry yeast. Allow the mixture to sit so the yeast can bloom, 10 minutes.
Add the remaining steamed bun ingredients to the bowl and, using the dough hook attachment, knead the dough with the motor running just above a stir, 8 to 10 minutes. Shape the dough into a ball and transfer to a greased bowl. Cover the bowl with plastic wrap and let the dough proof in a warm place until it has doubled in size, 1 hour and 15 minutes.
While the dough is proofing, make the pickles: Using a mandoline, slice the cucumbers into ⅛-inch-thick slices. In a medium bowl, toss the cucumbers with the sugar and salt, and let sit for at least 10 minutes, then set aside for later.
Make the BBQ mushrooms: In a medium bowl, combine the mirin, sake, soy sauce and liquid smoke. Set aside for later.
In a small saucepan, add the dried mushrooms and cover completely with water. Cover the mushrooms with a small plate so that they are completely submerged in the water. Bring the water to a boil, reduce the heat and simmer until the mushrooms are completely rehydrated, 5 minutes. Drain the mushrooms and let cool at room temperature. When the mushrooms are cool enough to handle, thinly slice them.
In a large sauté pan over high heat, add the olive oil and the sliced mushrooms. Stir the mushrooms a few times to ensure caramelization on both sides and cook for 5 to 7 minutes. Pour the soy sauce mixture over the mushrooms and cook until the liquid has reduced completely, 5 to 7 minutes. Reserve for later.
Punch down the dough and transfer to a lightly floured surface. Portion the dough into 24 even handfuls and shape into balls. Cover the portioned dough balls with a lightly dampened towel and let the dough rest again, 30 minutes.
Using a rolling pin, roll out the dough balls into 2-inch long ovals. Using your fingertips rub a small amount of vegetable shortening on the ovals, then fold the top over the bottom of each oval, hamburger-style. Shape all the balls into buns, cover once more with a dampened towel and let proof again, 30 minutes.
Set up a steamer basket and line with a piece of cut parchment paper. Be sure to leave some of the bottom uncovered so the steam can circulate. Steam the buns for 8 to 10 minutes, then serve with a smear of hoisin sauce, 3 to 4 pickles, a small handful of BBQ shiitake mushrooms and a garnish of green onions.