Heat a large skillet over low heat and add 1 Tablespoon of the oil. Add the onions with the pinch of salt and stir them to coat. Cover the skillet and cook the onions until softened and slightly golden, about 15 minutes, stirring every few minutes. Add the brown sugar and stir to combine, then over and cook for 5 minutes more. Add 1/2 Tablespoon of the oil to the skillet and add the mushrooms. Cover the skillet and cook the mushrooms, tossing once or twice, for 8 to 10 minutes. Stir in the garlic and cook for 30 seconds. Remove the mushrooms and garlic from the skillet, placing them on a plate.

Keep the same skillet hot over medium-low heat. To assemble the grilled cheese, brush the outside of each bread slice with the remaining 1 1/2 Tablespoons olive oil. Liberally spread the BBQ sauce on the inside of each bread slice. Evenly layer the cheese, onions, and mushroom slices on the bottom pieces of bread and top them with another handful of cheese. Add the top slices of bread.

Cook the sandwiches until golden brown on each side, 4 to 5 minutes.