Begin by cutting the salmon into four equal pieces, ensuring they are uniform in size for even cooking. Pat both sides dry with a paper towel, then season generously with salt and pepper.
Heat olive oil and butter in a skillet over medium-high heat until shimmering. Cook the salmon skin-side down for about five minutes until golden brown, then flip it over and cook for an additional 2-3 minutes until almost fully cooked through.
Carefully take the salmon out of the pan and place it on a plate while you prepare the sauce. If desired, you can remove the skin from each piece before returning it to the pan later.
In the same skillet, reduce excess oil while keeping about one tablespoon in there. Add chopped onion and sauté for around three minutes until lightly browned, then stir in minced garlic and red curry paste, cooking for about one minute until fragrant.
Pour in fish sauce if using, followed by coconut milk, brown sugar, matchstick carrots, and chopped broccolini into the skillet. Allow this mixture to gently bubble for about five minutes until vegetables are tender-crisp yet still vibrant.
Stir in lime juice before adding back in your seared salmon pieces so they can warm through without overcooking. Adjust seasoning as needed with more salt or pepper, then garnish with torn basil and cilantro before serving immediately over rice or noodles if desired.