Season the salmon on both sides with salt and pepper. In a 12-inch nonstick skillet over medium-high, heat the oil until shimmering. Add the salmon flesh side down, then immediately reduce to medium. Cook, undisturbed, until golden brown, 4 to 6 minutes. Using a wide, thin spatula, transfer the salmon browned-side up to a plate.

Return the skillet to medium-high and add the scallion whites and the garlic. Cook, stirring often, until lightly browned, 1 to 2 minutes. Add the tomatoes, carrots, curry powder and ½ teaspoon salt; cook, stirring, until the liquid released by the tomatoes has almost evaporated, about 5 minutes. Stir in the coconut milk, chilies and ¼ cup water, then bring to a simmer. Cover and cook over medium-low, stirring occasionally, until the carrots are softened, 6 to 9 minutes.

Stir in the lime juice and return the salmon, skin side down, to the pan. Cover, reduce to low and cook until the thickest parts of the fillets reach 120°F or are nearly opaque when cut into, 3 to 5 minutes. Off heat, taste the sauce and season with salt and pepper. Serve sprinkled with the scallion greens.