Prepare grill and heat to medium-high heat.
Heat 1 Tbs oil in a large skillet or wok over medium heat. Dice maitake mushroom and saute in oil until it starts to release its liquid and begins to brown, about 5 mins.
Add diced garlic scapes. Continue cooking until scapes change color and begin to soften (about 3 minutes for fresh garlic scapes. A little longer for frozen).
Reduce heat to low and stir in mirin and cooking sake. Cook until wines are completely absorbed, about 5 mins. Remove from heat and set aside.
Prepare the black garlic mayo by placing the black garlic cloves in a bowl and mashing them to a paste. Add mayonnaise and stir until thoroughly blended. An immersion blender will allow you to make things perfectly smooth, but mashing and stirring with a fork will get the job done quite well.
Grill your burgers to your preferred level of doneness (3-4 minutes per side for medium-rare). Remove from heat and allow them to rest on a plate for a few minutes.
While burgers are resting, butter the insides of your buns. Place them butter-side down on the grill to toast them (about 30 seconds).
Assemble the burgers by placing a pattie atop each lower bun. Top each with two shiso leaves and a few tablespoons of the maitake/garlic scape mixture. Spread a generous amount of the black garlic mayo on the inside of the top buns, then place atop the burgers, using the mayo to help hold the mushroom mixture in place. Serve immediately.