Combine the mayonnaise and 2 tablespoons pepperoncini brine in a small bowl. Toss the tomatoes with the chopped pepperoncini, the remaining 1 tablespoon brine, 2 tablespoons olive oil, half of the scallions and half of the mint in a medium bowl. Season with 1/4 teaspoon salt and a few grinds of pepper. Refrigerate the mayonnaise and the tomato salad while you make the burgers.
Combine the turkey, grated zucchini, feta, oregano, the remaining scallions and mint, 1/2 teaspoon salt and a few grinds of pepper in a large bowl and mix well with your hands. Form into 4 oblong patties, about 4 inches wide and 1/2 inch thick.
Heat the remaining 1 tablespoon olive oil in a large nonstick skillet over medium heat. Add the patties and cook until browned and cooked through, 4 to 5 minutes per side. Cut off a piece of each pita to open. Serve the burgers and zucchini sticks in the pitas and drizzle with the mayonnaise sauce. Serve with the tomato salad.