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Meanwhile, heat a grill, grill pan or cast-iron skillet over medium-high heat. Brush the tops of the portobello mushrooms with olive oil and season liberally with salt and pepper. Cook the mushrooms, stem side down, 5 to 7 minutes. Flip and continue cooking another 5 minutes. Remove to a work surface.
Fold the shredded cheeses into the caramelized onions just before building the burger. Carefully spoon the caramelized onions and melted cheese into the mushroom caps.
Build the burger by placing each stuffed mushroom on the bun bottoms. Top with the tomatoes and arugula. Smear the top buns with the Roasted Garlic Aioli and place on top of the other ingredients. Serve immediately.