1. Preheat the oven to 350 F. Scrape the tiny seeds from one half of the vanilla bean.
2. Beat the eggs with the vanilla seeds until frothy. Add the granulated sugar and beat with a whisk or hand or electric mixer until foamy and fairly thick.
3. Add the flour and continue to beat until thick and smooth. Add the cream, crème fraîche, or yogurt, milk, anise, and salt. Let rest while you prepare the pears and the baking dish.
4. Choose a 9 x 5 x 2-inch gratin dish or a 10-inch round deep pie plate or porcelain dish and smear it with the butter. Dust it with flour, rotating the pan so the flour sticks to all the butter, then inverting the dish to get rid of excess flour.
5. Cut the pears into sixths or eighths, remove the cores, and layer them attractively in the bottom of the dish. Sprinkle them with eau du vie, if you like. Pour the batter over, leaving just a little of the tops of the pears peeking through.
6. Bake for 20 minutes, or until the clafouti is nicely browned on top and a knife inserted into it comes out clean. Sift some confectioners' sugar over it and serve warm or at room temperature.