How to Make It
Heat grill to medium-high. Form the beef into four ¾-inch-thick patties. Use your fingers to make a shallow well in the top of each patty. (This will prevent overplumping during cooking.) Season with a total of ½ teaspoon each salt and pepper.
Oil the grill and cook the patties with the wells facing up for 4 to 5 minutes. Flip and cook until an instant-read thermometer inserted in the center registers 155° F for medium, 4 to 5 minutes more.
Meanwhile, brush the bread with the oil and grill until toasted, 1 to 2 minutes per side.
In a small bowl, mix together the yogurt with the Feta. Form sandwiches with the bread, burgers, Feta-yogurt sauce, tomato, and mint.