Cut the salmon into 4 equal pieces. Pat both sides dry with paper towel and season it with salt & pepper.
Add the oil and butter to a skillet over medium-high heat. Let the pan heat up for a few minutes. Prep your carrots and broccolini.
Cook the salmon, skin-side down, for 5 minutes. Flip it and cook for another 2-3 minutes (the fish should be almost done). If the fish is sticking at any point, give it a little longer to release naturally (don't force it).
Take the salmon out of the pan and transfer it to a plate. If you don't want to eat the salmon skin, you can easily slide it off before adding the salmon back into the pan.
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 (should be getting lightly browned), then stir in the curry paste and garlic and cook for 1 minute.
Add in the fish sauce, coconut milk, sugar, carrots, and broccolini. Let it gently bubble for about 5 minutes or so. The veggies should be tender-crisp, but you can cook them for a little longer if you prefer.
Stir in the lime juice, then add the fish back to the pan. Season with more salt & pepper as needed. Let it warm through for a couple of minutes. Sprinkle with the basil and cilantro and serve immediately.