In a large bowl whisk the marinade ingredients. Add the mushrooms and onion slices and turn to coat. Marinate at room temperature for at least 30 minutes or up to 1 hour, turning occasionally.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Grill the mushroom caps, gill sides down first, over direct medium heat, with the lid closed as much as possible, until cooked through, 8 to 12 minutes, turning once and brushing occasionally with the marinade. At the same time, grill the onion slices until softened and beginning to char, 6 to 8 minutes, turning once.
Chop the mushrooms and onions and place in a medium bowl. Gently stir in the bacon, thyme, parsley, and cheese. Season with salt and pepper.
Lightly brush the steaks with oil and season with salt and pepper. Allow the steaks to stand at room temperature for 15 to 30 minutes before grilling.
Increase the temperature of the grill to high heat (450° to 550°F). Grill the steaks over direct high heat, with the lid closed as much as possible, until cooked to your desired doneness, 6 to 8 minutes for medium rare, turning once or twice (if flare-ups occur, move the steaks temporarily over indirect heat). Remove from the grill and let rest for 3 to 5 minutes. Serve warm with the relish.
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