Pulse chicken in a food processor until coarsely chopped and transfer to a large bowl. Add water chestnuts, scallions, and jalapeño to processor and pulse until finely chopped, then add to chicken along with cilantro and salt. Stir together with your hands until just combined.

Form mixture into 18 (2-inch-diameter) patties on a baking sheet, then thread 3 patties through their sides onto each skewer.

Heat 1 teaspoon oil in a 12-inch nonstick skillet over moderate heat until hot but not smoking, then transfer 3 skewers of patties using a large metal spatula to skillet and cook until golden and just cooked through, about 3 minutes on each side. Transfer to a platter and keep warm, covered with foil. Add remaining teaspoon oil to skillet and cook remaining patties in same manner.