Remove the stems from the shiitake mushrooms. Either discard the stems or save them for a future stock.
In a bowl, whisk to combine the soy sauce, mirin, sugar, and sake. Pour the mixture into a skillet and add the shiitakes. Place the skillet over medium-high heat and bring to a boil. Reduce to a simmer, cover, and let braise for 15 minutes.
Once the 15 minutes are up, remove the mushrooms and set aside. Increase the heat to medium and reduce the soy sauce mixture until it reaches a syrupy consistency - about 10 minutes more.
While the shiitakes are braising, place the barley in a saucepan. Add 3 cups of water and 1/2 teaspoon of salt, if desired. Place the pot over medium-high heat and bring to a boil. Reduce to a simmer and let cook until the barley is tender and the liquid is mostly absorbed. This should take about 25 minutes. Drain the barley if any liquid remains and set it aside to cool.
With the barley and the shiitakes underway, you can work on a few of the toppings. Place the white onion in a small bowl and add the salt. Toss to coat. Set the onion aside.
Place the mayo and wasabi paste in a small bowl and stir to combine. Cover and chill until ready to serve.
Once the barley and the shiitakes are cooked and cooling, make the chickpea puree. Place the chickpeas and miso paste in a large food processor. Blitz until the mixture resembles mulch. Turn the food processor on and stream in the water, followed by the olive oil and sesame oil. Blitz until the mixture is smooth and velvety in texture.
Place the barley in a large bowl. Coarsely chop the shiitake mushrooms and add them to the bowl. Add the chickpea puree, panko breadcrumbs, egg, and scallions. Using a clean hand, mix until the ingredients are evenly dispersed and fully integrated.
Form the mixture into 8 patties, using a half-cup measure as a guide. Heat 2 tablespoons of neutral oil in a cast-iron skillet or griddle and place over medium heat. Once the oil is shimmering, add the patties 3-4 at time and sear until golden on both sides - this should take about 3 minutes per side.
To build your burger, split a burger bun and spread the wasabi mayo on the interior of the bottom half. Place the burger on top and drizzle with the reduced soy sauce mixture. Top the burger with pickled ginger, the white onion you set aside earlier, and a few leaves of romaine lettuce. Place the remaining half of the bun on top and serve immediately.