To prepare the burger buns, place the flours, yeast, sugar, salt and charcoal in a large bowl and stir to combine. In a separate bowl, whisk together the warm water, milk and aquafaba before adding it to the dry ingredients.
Stir with a large spoon until most of the mixture comes together before tipping the dough out on to a clean work surface. It will be floury, don't worry. Begin kneading the dough bringing together all the loose bits as you go. Knead for 3-4 minutes before adding a tablespoon of the coconut oil and working it through (I flour my hands because the dough get a little slippery here). Continue kneading for 2 minutes before adding the second tablespoon of coconut oil and a dash more flour if required. Knead the dough for another 5 minutes or until it is shiny and smooth.
Form the dough in to a ball and place it in a large, lightly greased bowl. Cover the bowl with cling wrap or a clean kitchen towel and set it in a warm room to proof for 1.5 - 2 hours and double in size.
When the dough has doubled in size, tip it out on to your work bench and knock it down. Divide it in to even size portions (I measured out 8 portions weighing 108 grams each) and roll them in to ball-shape buns. Place the buns on a lined baking tray with a good space between and rest them again, covered, for 45 minutes.
Preheat the oven to 205 degrees C (400 degrees F). Combine the maple syrup and dairy-free milk in a bowl and brush over each bun before placing them in the oven for 13 - 15 minutes. Remove the buns from the oven and brush with more glaze and set aside.
To prepare the mushrooms, heat the oven to 180 degrees C (356 degrees F). Combine the oil and Chinese five spice with the sea salt and whisk in a bowl. Lay the mushrooms on a lined roasting tray and brush with the oil mixture on both sides and set aside.
For the sprouts, remove the out layer and base of the sprout and finely slice the rest. Lay them out on a lined roasting tray and drizzle with tiny amount of oil and season with a little salt. Place both the mushrooms and the sprouts in the oven and roast for around 15 minutes, turning the mushrooms half-way through. When ready the mushrooms should be cooked through and the sprouts beginning to toast around the edges.
While the vegetables are roasting, whisk the wasabi, mirin and sunflower oil in a bowl.
Remove the vegetables from the oven and transfer the sprouts to a bowl while still warm. Add a few spoons of the wasabi oil and either massage it through or stir with a spoon. Taste and add more oil if needed.
To assemble the burger, cut the bun in half and add a good dollop of vegan mayonnaise to the base. Top with a mushroom followed by a small handful of the fresh green sprouts. Finish with a large spoon of the wasabi sprouts and top with the other bun half. Repeat with the remaining burgers.