Soak the raisins in the rum in a small bowl at room temperature for at least 4 hours or overnight, stirring occasionally.
Place the half-and-half, salt, and 1/4 cup sugar in a heavy, large saucepan. Scrape the seeds from the vanilla beans into the mixture, add the beans and bring to a boil over medium-high heat. Remove from the heat and let the vanilla beans steep for at least 1 hour or until the mixture cools to room temperature.
Beat the yolks and 1/2 cup sugar using an electric mixer until thick and pale lemon in color. Mix in the flour. Reheat the milk mixture and discard the vanilla beans. Gradually beat it into the yolks. Stir in the rum-soaked raisins. Return the mixture to the saucepan. Whisk over medium-high heat until the mixture comes to a boil and becomes very lumpy. Reduce heat to medium-low and whisk until smooth, for about 2 to 3 minutes.
Heat oven to 400 degrees F. Fill the pre-baked tart with the custard and smooth the top with a spatula. Brush with the egg. Transfer to oven and bake until the custard is well browned, 20 to 25 minutes. Cool in the tart pan on a rack for 3 to 4 hours.
To serve, remove the outer ring of the tart pan. Cut 6 to 8 wedges and transfer them to dessert plates. Stir the rum sauce over medium-high heat to reheat. Ladle over tart slices. Serve immediately.
In the bowl of an electric mixer fitted with the paddle attachment, cream the butter. Beat in the salt and sugar. Stir in the cream and 1/5 of the flour. Add the remaining flour, mixing with your fingertips or a fork until the ingredients are combined. Turn the dough out onto a piece of plastic wrap and shape into a flat disk. Wrap and refrigerate for at least 1 hour. (This can be prepared 1 day ahead.)
Butter a 10-inch tart pan with a removable bottom. Roll the dough out between 2 large sheets of plastic wrap into an 11-inch circle. Remove the top sheet of plastic. Invert into the prepared pan. Trim and finish the edges. Cover and refrigerate for at least 1 hour.
Heat the oven to 350 degrees F. Line the tart with parchment paper or aluminum foil and fill with beans or pie weights. Bake until the crust is firm and set, for about 20 minutes. Remove the beans and paper. Prick the bottom of the crust so it does not puff. Continue baking until well browned, 10 to 15 minutes.
Yield: 1 (10-inch) crust
Stir the brown sugar and heavy cream in a heavy small saucepan over medium heat until the sugar is melted. Pour in the rum and simmer, stirring until the sauce is smooth, thickened and reduced. Stir in the butter until melted. (This can be prepared ahead, cooled, covered and stored at cool room temperature for several hours or refrigerated overnight.)
Yield: 1 1/2 cups
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