- In a large skillet, brown the mushrooms in the butter. Add the shallots and sauté until golden brown. Season with salt and pepper. Add the garlic and thyme and cook for 1 minute. Deglaze with the broth, sauce, and vinegar. Bring to a boil and reduce until almost dry, about 2 minutes. Remove the thyme and adjust the seasoning. Keep warm.
- In a saucepan, off the heat, toss the cheese with the cornstarch. Add the milk and bring to a boil over medium heat, stirring with a whisk until the cheese has melted, about 5 minutes. Remove from the heat. Cover and set aside.
- Preheat the grill, setting the burners on high. Oil the grate.
- With your hands or a hamburger press, form eight large and thin patties with about 75 ml (1/3 cup) of meat each.
- Grill the patties for 2 to 3 minutes per side or until cooked through. Season with salt and pepper.
- Toast the buns. Generously spread the warm cheese sauce on the bottom buns. Add the patties. Top with mushrooms. Cover with a second patty and mushrooms. Top with cheese sauce, to taste, and arugula. Cover to close the burgers.
This cheese sauce is delicious with fries or nachos.