Place bread in a food processor and pulse into fine crumbs. Transfer to a large bowl. Pulse mushrooms in the food processor until finely chopped.
Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add onion, garlic and the mushrooms; cook, stirring occasionally, until tender and liquid has evaporated, about 10 minutes. Add to the breadcrumbs and let cool completely, 15 to 20 minutes.
Meanwhile, prepare Scallion-Lemon Mayonnaise, if using.
Preheat grill to medium-high.
Add ground turkey, egg, dill, mustard, salt and pepper to the mushroom mixture; mix well with a potato masher. With dampened hands, form the mixture into six 1/2-inch-thick patties, using about 1/2 cup for each.
Oil the grill rack. Brush the patties with the remaining 1 teaspoon oil. Grill until no longer pink in the center, about 5 minutes per side. (An instant-read thermometer inserted in the center should register 165 F.) Meanwhile, split buns and toast on the grill for 30 to 60 seconds, if using. Serve burgers on buns, garnished with lettuce, tomato and Scallion-Lemon Mayonnaise, if desired.
Ingredient note: Lean ground turkey breast has 110 calories and 1 gram fat per 3-ounce serving, compared to 193 calories and 11 grams fat in regular ground turkey (which may include leg meat and skin).