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Prepare the grill for direct cooking over high heat (450° to 550°F). Brush the steaks on both sides with oil and season evenly on both sides with salt and pepper. Let stand at room temperature while you prepare the mushroom sauce.
To make the mushroom-cognac sauce, warm the oil in a large (10- to 12-inch) skillet over medium-high heat on the stove. Add the mushrooms and sear until lightly browned, 3 to 5 minutes, stirring only once or twice.
Add the garlic and cook until fragrant, about 30 seconds, stirring occasionally. Turn off the exhaust fan, if using, and carefully add the Cognac (it may ignite, but it is fine if it doesn’t).
Simmer, stirring to scrape up any browned bits, until reduced by about half; it should take less than 1 minute. Stir in the broth and mustard, then add the cream and rosemary sprigs.
Bring to a robust simmer, then reduce the heat to medium-low and simmer gently until thickened to a saucy consistency, 5 to 8 minutes, stirring frequently. Discard the rosemary sprigs and stir in the salt and pepper. Set aside and keep warm. If the sauce becomes too thick, thin with broth, 1 tablespoon at a time.
Brush the cooking grates clean. Grill the steaks over direct high heat, with the lid closed, until cooked to your desired doneness, 8 to 10 minutes for medium rare (125° to 130°F), turning once or twice. Remove from the grill and let rest for a few minutes. Top the steaks with the sauce, garnish with the minced rosemary, and serve.