Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.
To make the aioli, combine the first six ingredients (mayonnaise, sour cream, minced garlic, minced ginger, freshly squeezed lime juice and salt) in a bowl and mix well. Reserve 2 tablespoons for the patties and cover and refrigerate the remainder until serving.
To make the glaze, combine the soy sauce, honey and rice vinegar in a small, heavy fire-proof saucepan. Mix the cornstarch and 1 tablespoon white wine in a small bowl until smooth and add to the soy mixture. Place on the grill rack and stir the mixture until it boils and thickens slightly, about 3 minutes. Set aside.
To make the patties, combine the reserved 2 tablespoons aioli with the egg, sour cream, lime juice and chile sauce in a large bowl and whisk to blend well. Add the chopped salmon, onion, cilantro, bread crumbs and salt. Handle the salmon as little as possible to avoid compacting it, mix well. Coat your hands with vegetable oil and divide the mixture into 4 equal portions and form the portions into patties to fit the buns.
When the grill is ready, brush the grill rack with vegetable oil. Place the patties on the rack, cover and cook until browned on the bottom, about 3 minutes. Turn the patties and brush the cooked side with soy glaze. Cook, turning and brushing frequently with the glaze, just until opaque throughout, 4 to 5 minutes longer. During the last few minutes of cooking, place the buns, cut side down, on the outer edges of the rack to toast lightly.
To assemble the burgers, top each bun bottom with an equal portion of the cucumber strips, a patty and an equal portion of the aioli. Add the bun tops and serve.