Cut three quarters of the cold salmon into 1/4-inch pieces. Finely mince the remaining salmon, then press and smear it with the side of the knife until slightly tacky.
Combine all chopped salmon with the egg, panko, scallions, dill, Dijon, lemon zest, garlic, smoked paprika, salt, and pepper. Fold gently just until evenly mixed.
Divide into four equal patties about 3/4 inch thick. Place on parchment, cover, and refrigerate for 20 minutes.
Whisk the mayonnaise, Greek yogurt, lemon juice, dill, parsley, grated garlic, salt, and pepper. Cover and refrigerate until serving.
Preheat an oiled grill or lightly oiled skillet over medium heat. Brush patties with olive oil. Cook for about 4 minutes on the first side and 3 to 4 minutes on the second side, turning once, until the centers reach 145°F (63°C).
Spread aioli on the toasted buns. Add lettuce, tomato, a salmon patty, red onion, cucumber, and more aioli as desired. Serve immediately.