Prepare grill.

Using clean hands, blend together the ground beef, kosher salt and pepper. Divide the mixture into four equal portions and shape into 1/2 inch patties. Set aside.

In a medium skillet, heat the olive oil over medium heat. Add the shallots and cook until the begin to caramelize, approximately 4-5 minutes. Add the mushrooms and cook for an additional 2-3 minutes. Stir in the garlic and cook until fragrant, approximately 1 minute. Add the wine to the pan and scrape the bottom of the pan with a wooden spoon to loosen any brown bits. Cook the mixture until reduced by 1/2, approximately 5 minutes. Stir in the thyme, Herbs de Provence and butter. Cook until thickened, approximately 2 minutes. Season with salt and pepper to taste.

In a small bowl, whisk together the creme fraiche, horseradish and chives. Season with salt and pepper to taste. Set aside.

Grill the burger patties over medium-high heat, 2-3 minutes on each side for medium rare. Top each burger with one slice of cheese. Turn off the heat and close the grill. Allow the cheese to melt, approximately 2-3 minutes.

Slice the buns in half and spread 1/4 of the horseradish sauce on the bottom of each bun. Place the burger on the bottom bun and spoon the mushroom sauce over top. Place the top of the bun on the burger and serve immediately.