Make the horseradish gremolata: Stir together all ingredients in a medium bowl. Set aside.

Make the salmon: Preheat oven to 325°F. Scrape the skin of salmon fillets with the back of a knife to remove any excess moisture, and pat dry. Place fillets, skin side down, on a paper towel–lined plate. Season the flesh with salt, and let stand for 15 minutes.

Heat 1/2 tablespoon oil in a large oven-safe skillet over medium-­high. Add lemons, cut sides down, and cook, undisturbed, until bottoms are charred, about 3 minutes. Transfer lemons to a plate. Scrape and discard any fond in bottom of skillet, and wipe clean.

Heat remaining 1 1/2 tablespoons oil in skillet over medium-high until shimmering. Add fillets, skin­ side down. Gently press on the fillets using a fish spatula, and move them around just a bit to ensure they do not stick. Cook until the skin begins to crisp, about 3 minutes. Transfer skillet to oven (without flipping the fillets), and bake until a thermometer inserted into thickest part of fillets registers 125°F for medium-rare, 6 to 8 minutes, or until desired degree of doneness.

Transfer fillets, skin side down, to a paper towel–lined plate to absorb any excess oil. Serve salmon, skin side up, with caramelized lemons and gremolata.