In a large mixing bowl, mix together the ground turkey, zaátar, salt, and pepper. Form the meat into 4 patties using your thumb to make an indent in the center of each patty.
Grill the burgers on a hot grill or grill pan for 5 minutes, or until lightly charred. Flip the burgers and immediately place the feta cheese on the burgers. Cook for 5 more minutes or until the internal temperature reaches 165 degrees.
Remove from the grill and set aside. Toast the english muffins and top the bottom half of each muffin with the arugula. Top with the burgers and then add the peppers and olives and the other half of the english muffin.