How to Make It
Bring wine, thyme, bay leaf, and peppercorns to a boil in a small saucepan over medium. Cook until mixture is reduced and almost dry, 3 to 4 minutes. Stir in cream, return to a boil, and cook until reduced by half, about 4 minutes. Reduce heat to low. Whisk in butter, 2 or 3 cubes at a time, waiting for each addition to melt before adding more. Season lightly with salt, and pour butter mixture through a fine wire-mesh strainer into a bowl. Discard solids. Keep beurre blanc warm.
Heat 1 tablespoon oil in a medium sauté pan over medium. Season both sides of fish with salt. Add fish, skin side down, to pan, lightly pressing on fish to flatten skin. Cook until skin is crisp, about 6 minutes. Flip fish, and cook until internal temperature of fish reaches 140°F, about 2 minutes. Remove fish from pan.
Stir together breadcrumbs and 1 1/2 teaspoons oil in a small bowl. Add breadcrumb mixture to pan, and cook, stirring constantly, until breadcrumbs are toasted, 3 to 4 minutes. Remove from pan. Increase heat to high, add remaining 1 1/2 tablespoons oil to pan, and heat until shimmering. Add mushrooms, and cook, undisturbed, until browned, about 90 seconds. Flip mushrooms, and add garlic. Cook until well browned, about 90 more seconds. Stir in breadcrumb mixture, and season with salt.
To serve, spoon about 3 tablespoons beurre blanc onto each of 2 plates; top evenly with fish, mushroom mixture, and chives.