Step 1
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.
Step 2
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.
Step 3
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.
Step 4
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.
Step 5
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.
