Check the fish for stray bones and slice into 4 pieces of equal size and thickness. Combine the carrot juice, wine, lime juice, shallots, ginger and 3/4 teaspoon kosher salt in a saucepan that's wide enough to hold the fish in a single layer. Bring the ingredients to a boil, then simmer over medium-low heat for 5 mins, whisking often. Place the fish in the carrot broth (it should be covered by the liquid) and cook very gently, adjusting the heat so that the broth never simmers, until the fish flakes apart when you bend it and is just at the point when it loses its last bit of translucence in the center, 7-10 mins.

While the fish is cooking, melt 1 tablespoon of the butter in a large skillet over medium heat. Add as much spinach as will fit, sprinkle with salt, and toss with tongs until it wilts. Add the rest of the spinach and cook until it is all wilted.

Put mounds of the spinach in the center of 4 warm shallow bowls and position a piece of the poached fish on top of each. Over medium heat, add the remaining 2 tablespoons butter to the broth, whisking vigorously until it is incorporated (or with an immersion blender if you own one). Stir in the cilantro and ladle the broth into the bowls. Serve right away.