Combine the Brussels sprouts, apples, vinegar, lemon juice, 1 tablespoon of the oil and some salt and pepper in a medium bowl. Toss to combine, and then fold in the crabmeat.
Combine the applesauce, pumpkin, 1/4 cup water and 2 tablespoons of the butter in a small saucepan. Cook over medium heat, whisking occasionally, until the butter has melted and the sauce is smooth and starting to simmer, about 5 minutes. Turn off the heat and whisk in the chocolate until melted. Season with salt and pepper. Cover with a lid to keep warm.
Sprinkle the halibut liberally on all sides with salt and pepper. Heat a medium cast-iron pan or nonstick skillet between medium and medium-high heat. When the pan is hot, add the remaining 2 tablespoons oil and swirl the pan to coat. Add the halibut top-side-down and cook until browned, 3 to 5 minutes. Carefully flip the fish. Add the remaining 2 tablespoons butter to the pan and cook until the other side is browned and the fish is just cooked through, tilting the pan and basting the fish with the butter, 3 to 5 minutes. Transfer to a plate.
To serve, spoon some of the sauce onto 2 serving plates. Spoon the slaw to one side of the plate and the fish on the other side. Serve immediately.