Stir together paprika, garlic powder, black pepper, onion powder, cayenne, oregano, and 1 teaspoon of the salt in a small bowl; set aside.

Heat 1 tablespoon of the oil in a medium-size nonstick skillet over medium. Add salmon; cook until lightly browned on both sides and sides of salmon are almost completely opaque but some pink remains in middle, 3 to 4 minutes per side. Transfer fillets to a large bowl, and set aside.

Wipe skillet clean. Add 2 tablespoons of the butter to skillet, and melt over medium. Add celery and white and light green scallion parts; cook, stirring occasionally, until softened, about 3 minutes. Add paprika mixture; cook, stirring constantly, until fragrant, about 1 minute. Transfer mixture to salmon in bowl. Add 1 tablespoon of the butter to skillet, and melt over medium. Add crushed crackers; cook, stirring and tossing occasionally, until toasted and dry, 2 to 3 minutes. Transfer to salmon mixture in bowl. Remove skillet from heat, and wipe clean.

Add 3 tablespoons of the mayonnaise and 1 tablespoon of the mustard to salmon mixture in bowl. Using a fork, stir mixture together, flaking some of the salmon. Continue stirring until salmon has broken down and mixture is chunky but bound. Shape mixture evenly into 4 (2 ½-inch-wide) patties; place on a large plate. Cover and refrigerate until patties are firm, at least 30 minutes or up to 2 hours.

Meanwhile, stir together chopped cornichons and brine, sugar, and remaining ½ cup mayonnaise and 2 teaspoons mustard in a small bowl; refrigerate until ready to use.

Place flour in a shallow dish; beat eggs in a separate shallow dish. Remove salmon patties from refrigerator, and place in flour; turn to lightly coat. Transfer patties to egg mixture, and turn to coat, letting excess drip back into bowl. Transfer patties back to flour, and turn to fully coat; tap off excess flour. Add remaining 2 tablespoons oil and 1 tablespoon butter to cleaned skillet; heat over medium. Add patties; cook until golden brown and crispy on both sides and cooked through, about 3 minutes per side. Transfer patties evenly to 4 plates or a platter; sprinkle evenly with remaining ¼ teaspoon salt. Sprinkle with reserved dark green scallion parts. Serve with pickle-mayonnaise sauce and lemon wedges.