Preheat grill on medium high heat.
To make the pickled red onions, bring the vinegars, water, and sugar to a boil in a pot and stir until the sugar has dissolved. Place the onions into a mason jar or heat proof container, and then pour the hot vinegar over the onions. Mix in the chili flakes, and cover the onions. Place in fridge to cool down.
To make the aioli, toast your dried shitake mushrooms in a pan over medium heat for 5 minutes until they are fragrant and golden brown. Transfer to a blender and pulse until they are very finely ground. Add in the rest of your ingredients and blend until smooth and all ingredients are well incorporated.
Cook bacon in a large sauté pan over medium heat for 5 minutes, until browned and crispy. Remove and reserve.
Drain off all but a teaspoon of the bacon fat, then return the pan to the stove. Add the butter and melt. Add the garlic and cook for 2 minutes until just beginning to brown. Add the mushrooms and continue cooking for 3 to 4 minutes, until the mushrooms are soft. Add the red wine and simmer until the liquid begins to cling to the vegetables. Fold in the reserved bacon. Season with salt and black pepper.
Season the patties on both sides with salt and pepper and cook for about 4 minutes, until nicely charred on the bottoms. Flip, immediately cover with the cheese, and continue cooking for another 4 minutes, until the cheese is melted and the burgers are firm but gently yielding to the touch.
Spread the mushroom aioli on both sides of the bun. Place a handful of arugula on the bottom of each bun and top with a burger. Divide the mushroom mixture among the 4 burgers, top with some pickled red onion and top with the bun tops.