Heat a deep-sided, large skillet on medium heat. Add 2 Tbsp neutral (canola or vegetable) oil to the skillet. Pat salmon fillets dry with a paper towel and place onto the hot skillet, skin side down. Let fry undisturbed for 3-4 minutes, or until the skin is browned and crispy. Flip salmon flesh side down and cook for no more than 30 seconds, allowing the flesh to just barely cook. Place salmon fillets onto a plate, skin side up, allowing the skin to stay crispy.
Add an extra 1 Tbsp of oil and the chopped broccoli to the same skillet and set to high heat. Leave the broccoli undisturbed on the skillet, letting it fry and char on the high heat, for about 5 minutes. Once the broccoli edges are charred, set aside on another plate. Add the diced onions to the same skillet with the reserved oil. Saute on medium heat for 2 minutes or until the onions are softened. Add the minced garlic and anchovies and continue frying for another 5 minutes, or until the garlic is fully cooked through. Add another drizzle of oil if the mixture is drying out or sticking to the skillet. Finally, add the tomato paste, sambal oelek, honey, rice vinegar, salt, and ground turmeric. Stir and saute for 7-10 minutes, or until the mixture has fried down into a deep brown paste. If the paste at any point sticks to the pan or burns, add a ¼ cup of water to deglaze the pan.
Add the coconut milk and water and stir to meld and incorporate the paste and liquids completely. Lower the heat to medium-low and add the charred broccoli and let simmer in the curry for 2-3 minutes or until just tender. Place the salmon fillets in the curry, skin side up so the pink flesh is submerged in the gravy. Simmer on low heat for 4-5 minutes, or until a fork can easily flake a piece of salmon off. Serve salmon curry with rice or warm naan.