Stuffed Cheesy Portobello Burgers

Name: Stuffed Cheesy Portobello Burgers
Ingredients: 2 tablespoons butter; 2 Vidalia onions, thinly sliced; 4 large portobello mushrooms, stems removed and caps cleaned; Olive oil, for brushing; Kosher salt and freshly cracked black pepper; 1 tablespoon brown sugar; 1 teaspoon balsamic vinegar; 1/2 cup shredded sharp white Cheddar; 1/2 cup shredded Swiss; 4 kaiser rolls, split, buttered and toasted; 4 slices tomato; 1 cup baby arugula; Roasted Garlic Aioli, recipe follows; 1 head garlic; 1 tablespoon olive oil; 1/2 cup mayonnaise; 1 teaspoon lemon juice