In a medium skillet, combine 2 tablespoons olive oil and red onion. Cook over medium heat until onions are beginning to soften. Season with salt and pepper and transfer to a roasting dish.
In the same skillet, combine tomatoes and balsamic vinegar. Cook over medium heat until the tomatoes are beginning to soften. Transfer to roasting dish with onions.
In a medium bowl, gently toss Portobello mushrooms with 1 tablespoon olive oil. Transfer to roasting dish with other ingredients. Add white wine.
Place roasting dish in oven and bake for 20-30 minutes. Remove from oven and let mixture cool to room temperature.
In a small saucepan, combine butter and thyme springs. Cook over low heat until butter is melted. Remove from heat and discard thyme.
Brush one side of each slice of bread with butter. Place slices buttered side down on a clean surface. Top half of the slices with 1/4 cup provolone cheese and 1/4 cup Monterey jack cheese. Top cheese with mushroom mixture, dividing evenly between the slices. Top with remaining bread slices.
Heat a griddle (or frying pan) to medium high heat. Place sandwiches on griddle to brown. Flip after 2 minutes to brown other side and cook until golden brown and cheese is melted