1 tablespoon extra virgin olive oil
7 tablespoons butter, at room temperature, divided
1 yellow or Spanish onion, thinly sliced
1 teaspoon sugar
Salt and pepper
1 tablespoon water
6-8 ounces mushrooms such as cremini, button, or shiitake, sliced 1/8" thick
1 sprig fresh rosemary, chopped
1 sprig fresh thyme, chopped
8 slices whole-grain bread
4 slices Wisconsin Cheddar cheese
4 slices Wisconsin Aged Brick cheese
4 large eggs
For caramelized onions: Heat large sauté pan over high heat; add olive oil and 1 tablespoon butter; heat. Add sliced onions and sugar. Cook without stirring for 2 minutes. Stir in pinch of salt and pepper. Cook, stirring at intervals, for another 5-8 minutes to brown onions. Add water and stir for 1 minute. Remove to a bowl and set aside.
Return pan to heat and add 1 tablespoon butter and melt; add mushrooms. Add pinch of salt and pepper and cook on medium-high heat for 3-5 minutes. Add chopped herbs and cook for 2 minutes. Remove to bowl.
Spread butter evenly on one side of the bread slices. Heat skillet or sauté pan over medium heat; place a bread slice, butter-side down, in skillet. Top with slice of Cheddar; add about 1 tablespoon each of mushrooms and onions, then top with slice of Aged Brick and another piece of bread, butter-side up. Grill for 2 minutes, then carefully flip over, cooking to a golden brown and melting the cheeses. Repeat with remaining ingredients (3 additional sandwiches). Place sandwiches on serving plates and return skillet to the heat. Add 1 tablespoon butter and fry eggs sunny side up over medium heat. Season with pinch of salt and pepper and fry until whites are just set. Place an egg on top of each sandwich and serve immediately.