The night or morning before you want to eat these skewers, stir together yogurt, lime juice, curry powder, cumin, salt, cayenne, and 1 tbsp. garlic in a large bowl. Transfer 3/4 cup spiced yogurt to a small container and chill until ready to use.
Stir remaining 1/2 tbsp. garlic into yogurt mixture in bowl. Add lamb pieces and stir to coat evenly. Cover and chill at least 8 hours and up to overnight.
Preheat a grill to medium-high (about 450°), then oil grate well with a wad of oiled paper towels. Remove lamb pieces from marinade (discard marinade) and thread onto six 12-in. skewers. Grill, uncovered, until charred outside and just cooked through (cut to test), about 5 minutes per side.
Sprinkle skewers with mint and serve with cucumber slices, pita, and reserved yogurt mixture (thinned with a little milk or water if you like).