Chop the garlic. Using the flat side of a chef’s knife, mash the garlic to a paste with a pinch of salt. Transfer the garlic paste to a small bowl and stir in the mayonnaise, chives, dill, mustard, lemon juice, cayenne, 1/4 teaspoons pepper, and salt to taste. Set the aïoli aside.
In a food processor, pulse the cubed bun into crumbs. Set aside 1/2 cup of the crumbs and save the rest for another use. Pulse the salmon until coarsely chopped, about 5 pulses. Transfer the salmon to a medium bowl and stir in 3/4 cup of the aïoli, the reserved breadcrumbs, 1/4 teaspoons salt, and 1/8 teaspoons pepper. Shape into four 1-inch-thick patties.
Heat 2 Tbs. of the butter in a 12-inch nonstick skillet over medium-high heat. When melted and hot (but not smoking or brown), add the patties and cook until they are firm and each side is crisp and golden-brown, 3 to 5 minutes per side.
Meanwhile, position a rack 6 inches from the broiler and heat the broiler on high. Melt the remaining 2 Tbs. butter in a microwave or on the stovetop. Brush it on the insides of the split buns. Put the buns on a baking sheet, butter side up, and toast under the broiler until light goldenbrown, 1 to 2 minutes.
Serve the burgers on the buns, spread with the remaining aïoli.
Serve with a deeply spiced, aromatic Chickpea and Spinach Salad.