Preheat oven broiler on high temperature (with the rack 3 inches from heat source) or grill on medium-high heat.
To prepare burgers, combine ground turkey, scallions, pepper, and egg, and mix well. Form into ½- to ¾-inch thick patties, and coat each lightly with olive oil.
Broil or grill burgers for about 7–9 minutes on each side (to a minimum internal temperature of 160 °F).
Combine mayonnaise and mustard to make a spread.
Assemble ¾ tablespoon spread, 1 ounce spinach or arugula, several slices of grilled portabella mushroom (optional), and one burger on each bun.