Remove fish from the refrigerator an hour before cooking to bring it close to room temperature. Pat both sides of fillets dry with a paper towel. Season flesh side with salt.
Heat oil in a large nonstick skillet over medium-high. Add fish, skin-side down, to pan, pressing firmly with a fish spatula to avoid the ends curling up. Cook 3 to 5 minutes, depending on thickness, until the flesh turns from translucent to opaque all the way up the sides. Remove skillet from heat, flip the fish, and let the residual heat finish cooking the flesh side. Transfer fish to a plate and wipe skillet clean.
Return skillet to burner over medium heat and add butter. Once melted, whisk in maple syrup, mustard, and curry powder. Cook 2 minutes, stirring often, until mixture starts to bubble and thicken. Add salmon back to skillet; use a spoon to generously coat fish in maple-mustard glaze. If desired, garnish with fresh chopped chives; serve.