1. Preheat grill to 350° to 400° (medium-high) heat. Combine first 5 ingredients gently. Stir feta cheese, onion, and oregano into meat mixture. Shape mixture into 6 (5-inch) patties.
2. Grill, covered with grill lid, 6 to 7 minutes on each side or until a meat thermometer inserted into thickest portion registers 170°. Serve on Homemade Hamburger Buns. Top each burger with sliced pickled green tomatoes, lettuce, thinly sliced cucumber, a pinch of dried crushed red pepper, and a fresh dill sprig.