In a bowl, combine feta cheese, oregano, rosemary, 1/2 tsp. pepper, and the lemon zest. Add ground meat and mix gently. Form into four 1-in.-thick patties.
In a blender, whirl olives, dill, parsley, lemon juice, sugar, and remaining 1/4 tsp. pepper. Add 3 tbsp. olive oil, 1 tbsp. at a time, to make a smooth paste.
Heat a gas or charcoal grill to medium-hot (you can hold your hand 1 to 2 in. above cooking grate only 3 to 4 seconds). Grill burgers, turning once, until browned on both sides and cooked through (cut to test), 10 to 12 minutes total.
Split rolls and toast cut sides on the grill. Brush toasted sides with remaining 1 tbsp. olive oil and spread each with about 1 tsp. of the olive spread. Top with burgers, tomato, cucumber, red onion, and romaine.
Note: Nutritional analysis is per burger.