Preheat barbecue for medium-high heat. Melt butter in a large frypan over medium-high heat. Sautee the eschalot until transparent, then add mushrooms and cook, covered, until darkened, about 5 minutes. Remove mushrooms from pan and set aside. Deglaze the pan with 1/2 the beef stock and burgundy wine. Increase heat and bring to a boil. Allow the mixture to boil until it has reduced by 1/3.
Barbecue the sirloin tips to desired doneness. Season with salt, pepper and thyme if desired. Remove to a plate and set aside.
When the sauce is reduced, stir in the rest of the stock and garlic. Return to a boil and continue to cook for another 5 to 10 minutes. Sauce will be thin. Whisk in flour and cook until the sauce is the desired thickness. Taste and adjust seasoning if necessary. Stir in the mushrooms. Serve tips with mushroom sauce spooned over.