Heat oil in a large skillet over medium heat. Add onion and cook 20 minutes, stirring frequently, until caramelized and very tender. Stir in sugar, salt, and pepper. Add bourbon, scraping up all brown bits in bottom of skillet. Stir in 1 1/2 teaspoon butter until melted; keep warm. (Makes about 1 cup.) In a medium bowl, toss cheeses until well combined. Divide evenly into 4 portions; press each into a disk-like patty to fit the size of the bread slices. Spread 1/4 cup of the onion mixture on each of 4 slices of bread. Top each with a cheese patty and another slice of bread. Spread 1 1/2 teaspoon of the butter on 1 side of each sandwich. Heat a large nonstick skillet over medium-low heat. Place 2 sandwiches, butter side down, in skillet. Cook 3 to 4 minutes, until golden brown. While first side is cooking, spread 1 1/2 teaspoon of the butter on other side of each sandwich. Turn sandwiches and cook 3 to 4 minutes, until second side is crispy and golden brown and cheese has melted. Repeat with remaining 2 sandwiches.