In a medium saucepan add 1/2 cup water, beer, brown sugar, and red onions. Season with salt and pepper, to taste, and stir to combine. Bring to a boil then reduce to a simmer. Simmer, stirring occasionally for about 1 hour or until the mixture has thickened and has reduced to about 2 cups. Transfer to a bowl and set aside to cool completely.
In a large bowl mix together the ground beef, beer and grill seasoning until well incorporated. Reserve half of the mixture for Round 2 Meatloaf recipe. Shape the remaining beef into 4 equal portions and flatten each portion. Place 2 tablespoons of cheese in the center of each and wrap the edges around the cheese to form a ball. Flatten each ball into patties. Place on platter and cover. Refrigerate if not cooking immediately.
Set up grill or indoor grill pan for direct cooking over medium-high heat. When ready to start grilling, oil the grate. Place patties on the hot oiled grill and cook 4 to 5 minutes per side for medium. Serve over lettuce on toasted buns topped with red onion compote. (Reserve leftover compote for Round 2 Ale and Onion Meatloaf.)