Shape the ground beef into 6 patties and season both sides with salt and pepper to taste. Heat a large skillet over medium heat. Place the burgers in the skillet and cook for about 5 minutes per side or until heated through.
In another skillet, heat the butter and olive oil over medium/high heat. Add the garlic cloves and onion to the skillet and cook until fragrant, about 1-2 minutes. Stir frequently.
Add the sliced mushrooms to the pan and cook for about 5 minutes or until the mushrooms are browned, stirring occasionally.
Add the milk and Neufchatel cheese to the pan, making sure to stir until the Neufchatel is melted. After it is melted, reduce the heat to low and add the dried thyme, dijon mustard, black pepper, and salt. Stir until combined and let this mixture simmer for 5 minutes and then spoon it on top of the burgers.
