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.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.