In a bowl, using a fork, gently stir the turkey with the basil, scallions, parsley, anchovy, 1 teaspoon of salt, 1/2 teaspoon of pepper and the 1/3 cup of mayonnaise. Using 2 lids from 1-quart takeout containers, press one fourth of the turkey mixture into 4 perfect patties. Transfer to a plate. Repeat to form the remaining patties.

Light a grill or preheat a grill pan. Season the patties lightly with salt and pepper. Grill over moderately high heat, turning once, until cooked through, about 8 minutes. Transfer the burgers to the buns, top with mayonnaise, onion and greens and serve.