To marinate the lamb, whisk together 4 tablespoons of the olive oil, the garlic and oregano in a medium baking dish. Add the lamb and turn to coat. Cover and let marinate in the refrigerator for at least 1 hour and up to 24 hours.
Heat your grill to high.
Remove the lamb from the marinade and season with salt and pepper on both sides. Place on the grill and grill until golden brown and slightly charred, 3 to 4 minutes. Turn the lamb over and continue grilling to medium-rare, 2 to 3 minutes longer. Remove to a cutting board, tent with foil, and let rest for 5 minutes before slicing into 1/2-inch-thick slices.
While the lamb is resting, brush each pita on both sides with the remaining 2 tablespoons oil and season with salt and pepper. Grill until golden brown and slightly crisp, approximately 30 seconds per side.
To make the salad, whisk together the lemon juice, vinegar, dill, salt and pepper to taste, and the honey in a large bowl. Slowly whisk in the extra-virgin olive oil until emulsified. Add the spinach, cucumber, tomatoes, olives and feta to the bowl and toss to combine.
Top each pita with some of the salad and top with several slices of the lamb.