Clean the mushroom caps gently with a damp paper towel. Avoid running them under water as they'll become waterlogged.
In a bowl, whisk together balsamic vinegar, olive oil, minced garlic, thyme, oregano, salt, and pepper.
Place mushrooms in a large zip-top bag and pour the marinade over them. Gently massage the marinade into the mushrooms. Let them marinate for 15-30 minutes, flipping once.
Preheat your grill to medium-high heat (around 400°F).
While the grill heats up, sauté the sliced onions in a pan with a bit of olive oil until caramelized, about 10-12 minutes.
Place mushrooms on the grill, gill side up. Grill for 5-6 minutes.
Flip the mushrooms and add cheese if using. Grill for another 4-5 minutes until the mushrooms are tender.
Toast the burger buns on the grill for 1-2 minutes.
Assemble your burgers: place the mushroom on the bottom bun, top with caramelized onions and fresh arugula.