For the relish Whisk oil, vinegar, lime juice, mustard, ginger, and salt in bowl until smooth. Stir in cucumber, mint, cilantro, and chile. Season with salt and pepper to taste. Set aside for serving.
For the salmon Dissolve ¼ cup salt in 1 quart water in large container. Submerge salmon in brine and let sit at room temperature for 15 minutes. Remove salmon from brine and pat dry with paper towels.
Sprinkle bottom of 10-inch nonstick skillet evenly with ¼ teaspoon salt and ¼ teaspoon pepper. Place fillets, skin side down, in skillet and sprinkle tops of fillets with remaining ⅛ teaspoon salt and remaining ⅛ teaspoon pepper. Heat skillet over medium-high heat and cook fillets without moving them until fat begins to render, skin begins to brown, and bottom ¼ inch of fillets turns opaque, 6 to 8 minutes.
Using tongs, flip fillets and continue to cook without moving them until centers are still translucent when checked with tip of paring knife and register 125 degrees (for medium-rare), 6 to 8 minutes. Transfer fillets, skin side down, to plate and let rest for 5 minutes before serving with cucumber-ginger relish.