1 tablespoon chopped dill plus 2 sprigs for garnish
2 large sole fillets without skin
3-4 tablespoons dry white vermouth
2 sheets Parchment paper
Preheat oven to 375 degrees
Melt 2 tablespoons of butter in a frying pan, add fennel, salt, and pepper. Cook over medium heat for 5 to 8 minutes, until the fennel is lightly browned but still fairly crisp. Stir in the dil, taste, and adjust seasoning.
To shape the paper, fold each of parchment paper in half lengthwise and cut so that when the paper opened flat it forms a heart shape. Melt the remaining 2 tablespoons of butter and brush the paper with it.
Spread the completely cool fennel on one side of the fold, leaving a 3-inch border of paper. Set fish on top of fennel. Sprinkle the fennel with with the vermouth, salt, and pepper and add the dill sprig. Fold the paper back over the sole and press the edges together. Starting at the curved side, pleat the paper on itself to seal the edges, working around to finish at the point with a quick twist of paper. You can prepare the paper cases up to 2 hours ahead.
Transfer to a baking sheet and bake until they are puffed and brown, 15 to 20 minutes. Steam from the fish and fennel during cooking swells the paper into a balloon. Transfer the packages to warm plates and serve at once.