Heat grill to medium-high. In a medium bowl, place chicken, scallions, ginger, lemon juice, paprika, cumin, cardamom, cayenne, 1 1/2 teaspoons coarse salt, and 1/2 teaspoon pepper; toss to combine. Set aside to marinate, at least 10 and up to 30 minutes.
Transfer chicken mixture to a food processor; pulse until roughly chopped, but not pasty, 10 to 12 times. Gently form mixture into sixteen 3/4-inch-thick patties (about 3 tablespoons each).
Moisten a folded paper towel with oil; grasp with tongs and rub over grates. Season patties with salt and pepper; grill until opaque throughout, 2 to 3 minutes per side.
For the Cumin Yogurt Sauce: In a small bowl, combine 1/2 cup plain low-fat yogurt with 1/2 teaspoon ground cumin; season with coarse salt and ground pepper.
Halve pitas crosswise (toast on grill, if you like). Into each pocket, place 2 chicken patties, cucumber slices, and cilantro sprigs. Serve with Cumin Yogurt Sauce and, if desired, watermelon slices.