Vegetarian BLT With Shiitake Bacon

Name: Vegetarian BLT With Shiitake Bacon
Ingredients: 14 ounces fresh shiitake mushrooms, stems removed, caps halved if large; 1 ½ tablespoons olive oil; 1 ½ tablespoons tamari; ½ teaspoon fine sea salt; ¼ teaspoon freshly ground black pepper; 2 to 4 tablespoons mayonnaise; 4 whole-grain bread slices, lightly toasted; 4 green leaf lettuce leaves (from 1 head); 6 tomato slices (from 1 medium [6 ounce] tomato)