Heat 1 Tbsp. vegetable oil in a large nonstick skillet over medium-high. Season four 6-oz. boneless salmon fillets, preferably skin-on, with kosher salt and cook (skin side down if it has skin), pressing fillets gently with a spatula, until golden brown, about 4 minutes. Turn fillets over and cook until golden brown and just cooked through, about 3 minutes. Transfer to a plate. Wipe out skillet and reserve.
Purée 1 medium onion, coarsely chopped, 2–3 green Thai chiles, depending on heat preference, 3 garlic cloves, one 1½” piece ginger, peeled, 1 Tbsp. double-concentrated tomato paste, ½ tsp. ground turmeric, ¼ cup cilantro leaves with tender stems, and one 13.5-oz. can unsweetened coconut milk in a blender on high speed until smooth. Pour onion mixture into reserved skillet and sprinkle with 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt. Bring to a brisk simmer over medium-high heat, then reduce heat to medium and cook, stirring often with a wooden spoon, until mixture is reduced to a thick paste, fat separates from coconut milk (it will look broken), and aromatics are golden brown, 30–35 minutes.
Pour in remaining one 13.5-oz. can unsweetened coconut milk and ¼ cup water and bring to a simmer, scraping up any browned bits.
Gently break salmon fillets into 3″ pieces and return to skillet. Cook just until salmon is heated through, about 3 minutes. Stir 1 Tbsp. fresh lime juice into curry and remove from heat.
Divide curry among shallow bowls and top with cilantro leaves with tender stems and halved green Thai chiles. Serve with cooked basmati rice or country-style bread and lime wedges alongside.
Do ahead: Curry base can be made 1 day ahead. Let cool; cover and chill. Reheat over medium-low, adding water to thin as needed.