For the caramelized onions
Add the butter to a large pan over medium-high heat followed by the onions. Stir frequently to insure even color. Once they start to caramelize, deglaze with some champagne to help soften them and add some sweet champagne flavor. Reduce, let the onions caramelize more and repeat until the onions are well caramelized and dark brown. Set aside.
For the Champagnnaise
Add the egg yolk and dijon mustard to a medium bowl and, while whisking, slowly emulsify the oil a little bit at a time. Once all the oil is incorporated and you have a thick emulsion add the other ingredient and fold in.
For the rest
Salt and sear the burger pattys on a griddle or pan over high heat melting the cheese and grilled onions to the top in the last few minutes of cooking.
Spread butter on the inside of the brioche buns and toast in the pan or under a broiler. Spread each side of all four buns with a generous amount of champagnnaise. Assemble with the patty, cheese, and onions and serve.