1. Prepare poaching liquid: Gently squeeze kumquats over a measuring cup to release juice. Melt butter in a large skillet over medium heat. When it is foamy, sweat the shallot until it is softened but not taking on any color, about 2 minutes. Add wine, tangerine juice, and kumquats (and juice) and bring to a simmer.

2. Poach fish: Fold thin ends ("tails") of fish fillets under a little so the fillets have a more uniform thickness. Season with salt, then gently slide the fillets into the simmering liquid. The fish will not be completely covered with liquid but should be in a single layer. Cover with a parchment-paper round and reduce heat so the liquid is barely moving (just under a simmer). Poach until fish is opaque throughout and flaky to the touch. (Check after 1 minute, but it may take up to 5 minutes.) Use a slotted fish spatula (or other long spatula) to transfer fish to a platter. Cover with parchment paper and set in a warm spot until ready to serve.

3. Make buerre blanc: Pour the cooking liquid through a fine sieve to strain out the solids, pressing on the fruit to extract as much liquid as possible. Return liquid to skillet and boil over moderate heat until reduced to about 3 tablespoons, 2 to 3 minutes. Turn off heat and gradually add the butter, whisking in a few cubes at a time. If butter isn't melting, return pan to moderate heat and swirl to hasten melting. Whisk until each addition is incorporated before adding more. When all of the butter has been added, stir in the sliced kumquats. You should have about 1/2 cup sauce. Season with salt and pepper.

4. Serve: Drizzle buerre blanc over fish, and garnish with mint.