Preheat the oven to 180
Mix extra virgin olive oil, tamari and black pepper- rub onto mushroom caps
Place cap side down on a rimmed baking sheet and bake for about 15 minutes
Cool to room temperature; set aside.
Add oil, onions and fennel to pain on medium head
Cook until onions are soft, about 4-5 minutes.
Add the sugar and saute 1 minute
Add the red wine and balsamic, cover and simmer 10 minutes
Remove the lid and cook until most of the liquid evaporates
Remove from the heat and season to taste with salt and pepper
Whisk together mayonnaise, mustard, garlic, oregano, parsley, lemon zest and olive oil
Refrigerate until ready to use
Slice the mushroom caps, arrange on baking tray and sprinkle with ground pepper.
Place 2 to 3 tablespoons of the caramelized onions on top.
Bake for 5 to 7 minutes.
Meanwhile, warm the rolls in the oven.
Assemble burger with aioli, mushroom, onion, cheese and salad