Pat the salmon fillets dry with paper towels and season them generously with salt and pepper on both sides.
Heat a large skillet over medium-high heat and add the butter and olive oil.
Once the butter has melted and the skillet is hot, carefully place the salmon fillets skin-side down in the skillet.
Cook the salmon for 4-5 minutes on the skin side, without moving them, until the skin is crispy and golden brown.
Using a spatula, carefully flip the salmon fillets over and continue cooking for another 2-3 minutes until the salmon is cooked through, but still moist and tender in the center.
TO STORE. Store leftover cooked salmon in an airtight container in the fridge. Eat it within 3 days. TO REHEAT. You can bring the salmon to room temperature and use it cold for salads and sandwiches. If you want to warm it up, use the microwave or put the fish in the oven preheated to 275F degrees for 15 minutes or until heated through. TO FREEZE. Put cooked salmon fillets into a freezer bag, push out the excess air, and freeze for up to 3 months. Defrost the fillets in the fridge before reheating them.