Heat grill to medium-high. In a small bowl, combine the yogurt, cucumber, mint, and ¼ teaspoon salt. Set aside.
In a large bowl, combine the lamb, shallot, garlic, thyme, ½ teaspoon salt, and 1/8 teaspoon pepper. Form 4 patties about ¾ inch thick.
Grill the patties until cooked through, 5 to 6 minutes per side. (Or heat the oil in a grill pan over medium heat and cook patties, 5 to 6 minutes per side.)
Serve patties in the pita pockets with the cucumber relish.