Place a 4QT Dutch oven on medium-high heat. Melt a tablespoon of butter in the pot and add in the sliced mushrooms. Cook fully and then remove from the Dutch oven.
Add three tablespoons of butter with olive oil to the Dutch oven. Place all of the sliced onions in the Dutch oven and cook on medium-high, stirring constantly. Cook so that the onions start to loose all their water content, soften and reduce.
Once the onions lose their water content, reduce the heat to a medium heat. Stir every minute or so for 30-40 minutes, until the onion have a nice caramelized color.
Add the minced garlic, thyme, and cooked mushrooms to the caramelized onions. Cook for 2 minutes while stirring.
Deglaze the Dutch oven with the wine. Scrape the bottom of the pot to release all of the stuck on flavors.
Add the beef broth and the brewed coffee to the Dutch oven. Bring to a simmer and cook for 10 minutes.
Pour the soup into oven safe bowls, placing them on a baking tray. Place the prepared crostini to cover the surface of the bowl.
Sprinkle the grated cheese over the crostini. Carefully place the tray with soups under the oven broiler to crisp up the cheese.