Heat a grill pan or outdoor grill to medium-high heat. Drizzle olive oil evenly over mushroom caps liberally then rub with rosemary and grill seasoning. Grill 3 to 4 minutes on each side - until dark and tender. Remove the caps and reserve.
While mushrooms cook, toast split rolls or bread under broiler until deeply golden then rub with 1 clove of cracked, split garlic. Drizzle extra-virgin olive oil over the bread and season with a little pepper then chop bread into cubes.
Rub the inside of a bowl with remaining cracked clove of garlic. Combine the lemon juice, Dijon, Worcestershire, hot sauce and anchovy paste. Whisk in 1/3 cup extra-virgin olive oil. Add the greens and bread to the bowl and toss greens with dressing to coat. Add cheese to salad and toss again then season with black pepper, to taste. Top servings of salad with sliced mushrooms across the top.