Preheat oven to 400 degrees. Preheat an ovenproof skillet large enough to accommodate both fillets over high heat. Dredge fillets in flour, shaking off excess. Add olive oil to skillet and add fillets, skin side up. Cook until golden brown on one side. Turn over and add shiitake mushrooms, turning to cook evenly. When mushrooms are slightly wilted, add shallot and season with salt and pepper.
Remove skillet from heat and deglaze pan with the Cognac. Add currants. Once Cognac has stopped sizzling, return skillet to the heat and cook until Cognac is almost entirely reduced. Add chicken stock. Place skillet in the oven for about 8 to 10 minutes, or until fish is tender.
Place each fillet on a plate and return the skillet to high heat. Reduce sauce to half its original volume and add butter, stirring to incorporate. When butter is completely melted and blended into sauce, spoon over each fillet and serve.