In a large skillet, melt the butter over medium heat. Add the onions and cook just until translucent, about 5 minutes. Add the brown sugar and balsamic vinegar, cover and continue cooking, stirring occasionally, until the onions are caramelized, 20 to 25 minutes.
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.
Roasted Garlic Aioli:
Preheat the oven to 400 degrees F.
Slice off the top of the garlic head. Drizzle the cut side with the olive oil and sprinkle with salt and pepper. Wrap in foil and bake until tender and fragrant, 35 to 40 minutes. Allow to cool.
Remove the garlic cloves from their skins and add to a food processor along with the mayonnaise, lemon juice and a pinch of salt and pepper. Pulse until pureed. Season and store in an airtight container in the refrigerator until ready to use.