Combine the cider, apples, 2 shallots, 2 tablespoons vinegar, 1 teaspoon curry powder, and salt and pepper to taste in a saucepan and bring to a boil. Cook until the liquid evaporates, about 12 minutes; let cool slightly. Stir in 2 tablespoons chopped cilantro.
Meanwhile, pulse three-quarters of the salmon in a food processor until finely ground. Add the remaining salmon; pulse until chopped. Transfer to a bowl; stir in the panko, mayonnaise and the remaining shallot, 2 tablespoons cilantro, 1 tablespoon vinegar and 2 teaspoons curry powder. Add salt and pepper. Form into four 1/2-inch-thick patties.
Melt the butter in a large nonstick skillet over medium-high heat. Add the patties and cook until browned and cooked through, 2 to 3 minutes per side. Butter the buns and sandwich with the salmon burgers, mayonnaise, the apple relish and more cilantro. Serve with chips.