Season Salmon fillets on both sides with salt and pepper.
Heat 2 tablespoons of oil in a skillet. Place Salmon fillets skin side down and sear for 3-4 minutes without moving. Carefully flip and cook for another 4 minutes. Remove salmon from the skillet and keep aside covered.
To the same skillet, add rest of the oil. Saute onion for 2-3 minutes until soft. Then add minced garlic and ginger. Saute on low heat for about a minute, until the raw smell is gone.
Stir in the red curry paste and saute for 1 minute. Add a splash of water if the mixture is getting stuck to the skillet.
Pour coconut milk into the skillet and mix well until the mixture is creamy and no lump of red curry paste in the mix.
Then add brown sugar, salt, and pepper. Simmer for 1-2 minutes.
Season with Soy sauce, fish sauce, and lime juice. Add crushed Kaffir lime leaves and slit bird-eye chili if using. Give everything a good mix.
Carefully place the cooked salmon fillets back to the skillet. Pour of the curry sauce on the fillets and switch off the flame.
Sprinkle fresh cilantro on top and serve.