Special equipment: 6 (8-inch) wooden skewers
Add water chestnuts, scallions, and jalapeno to a food processor and pulse until finely chopped. Transfer to a bowl with the 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.