Here we make a quick aioli by combining two pantry staples: mayonnaise and roasted red peppers. Stirring a bit of the sauce into the burger mix helps the patties hold together nicely and stay moist.
Combine vinegar, water, 1 tablespoon dill, crushed garlic and salt in a small saucepan. Heat over medium-high heat until the salt dissolves, about 2 minutes. Remove from heat. Thinly slice cucumber into ribbons using a vegetable peeler. Stir the cucumber into the brine and set aside.
Combine mayonnaise, roasted red peppers and grated garlic in a small bowl and stir well. Transfer 1/4 cup of the aioli to a large bowl and set the rest aside. Stir salmon, chives and the remaining 1 tablespoon dill into the aioli in the large bowl. Form the mixture into 4 patties about 4 inches in diameter.
Heat oil in a large nonstick skillet over medium-high heat. Add the patties and cook, flipping once, until golden and an instant-read thermometer inserted in the center registers 145°F, 6 to 8 minutes total.
Drain the pickles. Serve the burgers on buns with arugula, the remaining aioli and the pickles.