In a medium bowl, combine lamb, curry powder, and 2 sliced scallions; season generously with salt and pepper. Mix gently with a fork (do not overmix); form into 4 oval patties to fit on bread (each 1 inch thick).
In a large skillet, heat oil over medium-high. Season patties with salt and pepper. Cook until browned and just cooked through, 4 to 6 minutes per side (reduce heat if browning too quickly).
Meanwhile, in a small bowl, mix feta, yogurt, and remaining scallion, mashing cheese gently to combine; season with salt and pepper.
Spread 4 slices of bread with feta mixture; top with cucumbers, lamb patties, and remaining bread.
Vegetable peelers can slice as well as peel; use one along the length of a Kirby. The ribbons will stay put on a burger and ensure some crunch in each bite.