In a medium bowl, mix the ground lamb, 2 tbsp. EVOO, oregano, garlic and cumin. Form into 4 patties (thinner in the centers for even cooking). Transfer to a plate; cover and refrigerate while preparing the marinated cucumbers.
In another medium bowl, mix the cucumbers, onion, vinegar, dill and 2 tbsp. EVOO. Season with salt and pepper; cover and refrigerate.
Heat a griddle pan, a large, heavy skillet or a grill over high. Season the patties; cook, 4 to 5 minutes per side for medium. Drizzle the cut sides of the ciabatta rolls with EVOO and cook, cut sides down, until charred in spots, about 2 minutes.
Divide the patties among the roll bottoms. Add a tomato slice, some tzatziki, some of the marinated cucumber mixture and the roll tops. Serve the burgers with the remaining marinated cucumbers on the side.