1. Heat 1 tablespoon olive oil and 1 tablespoon butter in a medium high-sided skillet over medium-high heat. Add 8 ounces mushrooms and cook, stirring, until lightly browned, 1 to 2 minutes. Season with salt and espelette pepper. Add sake to deglaze, scraping up any browned bits from bottom of skillet; cook until sake is reduced by half.

2. Add chicken stock and 1/8 teaspoon lemon juice; season with salt and espelette pepper. Add rosemary sprigs and let cook until liquid is reduced by half. Using a slotted spoon remove mushrooms and rosemary; set mushrooms aside and discard rosemary.

3. Gradually add remaining 1/4 cup (1/2 stick) butter and 1 tablespoon olive oil to skillet, using an immersion blender to combine. Add mushrooms and stir to coat. Set mushrooms aside in a warm place.

4. Preheat broiler; lightly oil a small baking sheet and set aside. Heat remaining 2 tablespoons olive oil in a large nonstick ovenproof pan over high heat. Season on both sides with salt and pepper. Add fish to skillet, skin side down. Cook until skin is crisp, 4 to 5 minutes. Transfer to prepared baking sheet and place under broiler; cook until fish is no longer opaque, about 1 minute more. Remove from broiler and keep warm.

5. Add chopped rosemary and stir into mushroom mixture along with chives; divide mushroom mixture evenly among 4 plates. Top each with a fish fillet and thinly sliced mushrooms; season with salt and pepper. Spoon any remaining sauce from mushroom mixture around fish; serve immediately.