Make the caramelized onions. Heat 1 tablespoon olive oil in a saute pan over medium heat. Add the onions and saute until softened, about 10 minutes. Reduce the heat to low and season the onions with ½ teaspoon salt, and cook another 10 minutes, stirring occasionally. Add ¼ cup bourbon and simmer until all of the alcohol has cooked off. Continue to cook the onions until deep brown and caramelized.
Prepare the rosemary honey. Combine ¼ honey with 1 sprig rosemary in a small saucepot. Bring to a simmer over medium heat. Reduce to low and continue to simmer for 10 minutes. Remove from heat and let sit for 10 more minutes. Remove the rosemary and set aside.
Spread the softened butter on one side of each of the bread slices. Flip the bread over and place the caramelized onions and shredded fontina on 2 of the bread slices. Top with the remaining bread.
Heat a nonstick saute pan over medium-low heat. Carefully lay the sandwiches on the pan and cook until golden brown., about 3 minutes. Flip the sandwich and cook for another 2-3 minutes or until golden brown and the cheese has melted.
Drizzle the rosemary honey on top and serve immediately.