1. Sprinkle beef with 3/4 teaspoon salt and 1/4 teaspoon pepper; gently mix, and shape into 8 thin patties.
2. Place 1 piece feta cheese in center of each of 4 patties. Top with remaining 4 patties, pressing edges to seal.
3. Brush onion slices with 2 teaspoons oil, and sprinkle with the remaining 1/8 teaspoon salt and 1/8 teaspoon pepper. Cook on a lightly greased grill pan or in a skillet over medium-high heat 4 minutes on each side or until tender and slightly charred. Transfer to a plate, and keep warm. Add burgers to pan; cook 6 minutes on each side. Transfer to a plate, and keep warm. Brush bread slices with remaining 4 teaspoons oil. Add to pan, and toast about 1 minute on each side or until lightly golden.
4. Spread Dijon mustard evenly on 1 side of each bread slice. Top each with 1 onion slice, 3 tomato slices, and 1 burger. Serve immediately.