In a saucepan over medium heat, melt butter. Stir flour into mixture and cook for about 1 minute.
Whisk in milk and simmer for about 2 minutes to thicken a bit. Remove saucepan from heat and whisk in cheese, until melted and smooth. Stir in salt and cayenne.
Place chicken burgers on brioche rolls and spoon cheese sauce over them. Top with caramelized onions.
Add butter and 1 Tbsp olive oil to a large skillet over medium heat. When butter is melted, add onions.
Continue cooking for 20 minutes, stirring every 5 minutes or so, until onions are well caramelized. Stir in salt and pepper.