Pat the halibut fillets dry with paper towels, season with salt and pepper and set aside.
Heat the butter and the olive oil in a large pan over medium-high heat. Once the butter has melted and starts to brown a little, swirl the pan to combine the butter with the olive oil.
Add the halibut fillets, presentation side down, and sear for 3-4 minutes or until the edges of the fish begin to turn opaque and the fish easily releases from the pan.
Using a flat spatula, flip the fillets and add the lemon slices to the pan. Continue cooking for another 4-5 minutes or so, until the fish is just starting to flake and is opaque in the middle.
Transfer the seared halibut to a plate and top with salmon butter. Spoon some hot oil from the pan over salmon butter to let it melt a little. Garnish with pan-fried lemon slices, drizzle with lemon juice, sprinkle with chopped parsley and serve.