Cut the salmon into 4 equal pieces. Pat both sides dry with paper towel and season with salt & pepper.
Add the oil and butter to a skillet over medium-high heat. Let the pan heat up for a few minutes.
Cook the salmon, skin-side down, for 5 minutes. Flip it and cook for another 2-3 minutes.
Take the salmon out of the pan and transfer it to a plate.
Spoon some of the oil out of the skillet so you have about a tablespoon of it left in there.
Add the onion to the skillet and sauté for about 3 minutes, then stir in the garlic and curry paste and cook for 1 minute.
Add in the fish sauce, coconut milk, sugar, carrots, and broccolini. Let it gently bubble for about 5 minutes.
Stir in the lime juice, then add the fish back to the pan. Let it warm through for a couple of minutes. Season with more salt & pepper as needed.
Sprinkle with the basil and cilantro and serve immediately.