Heat olive oil in a large skillet over medium heat. Add onion and cook for 10 minutes or until softened and just beginning to turn golden brown, stirring frequently.
Add lager, salt, and pepper to skillet and increase heat to medium-high. Cook 5 minutes more, continuing to stir frequently, until liquid is absorbed. Remove from heat.
Divide onions onto two slices of bread and top with cheese and remaining bread slices. Butter bread on each side.
Wipe out skillet and return to medium-high heat. Place sandwiches in skillet and cook until browned, about 3β5 minutes on each side (depending on how toasted you like your grilled cheese).
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