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.