In a medium bowl, season the meat with salt and pepper. Mix in the spinach, herbs, onion and garlic. Add the feta and mix gently to just combine. Form the meat mixture into 4 patties (thinner in the center for even cooking).
In a medium nonstick skillet, heat a drizzle of EVOO over medium-high. Cook the patties, turning occasionally, until browned on both sides, 8 to 10 minutes for medium. Douse the burgers with the juice of half a lemon.
In another medium bowl, toss the lettuce, tomato, oregano and the juice of the remaining lemon half; season. Top the roll bottoms with some of the salad, the patties and the roll tops.