The Birchwood Grilled Cheese

Ingredients

• • 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