How to Make It
Light a grill. In a small saucepan, boil the broth over high heat until reduced to 1/2 cup, 7 minutes. Remove from the heat.
In a large skillet, heat 2 tablespoons of the oil. Add the chanterelles, season with salt and pepper and cook over moderately high heat until the liquid they release has evaporated, about 4 minutes. Continue cooking, stirring, until the mushrooms begin to brown, about 5 minutes longer. Add the remaining 4 tablespoons of oil, the bay leaves, shallots, garlic and thyme and cook over low heat, stirring, until the mushrooms are tender, 7 minutes. Add the reduced beef broth and simmer for 1 minute. Remove from the heat and stir in the vinegar and mustard. Season with salt and pepper and remove from the heat.
Season the steaks generously with salt and pepper and grill over high heat until nicely browned and an instant-read thermometer inserted in the center registers 130° for medium-rare, about 6 minutes per side. Transfer the steaks to a cutting board and let rest for 10 minutes. Reheat the chanterelles and discard the bay leaves and thyme sprig. Thickly slice the steaks and serve with the mushrooms.