To Make the Ice Cream
In a small, heavy saucepan, combine the water, ¼ cup of the sugar, and the 2 tablespoons rum. Cook over medium-low heat, stirring occasionally, until the sugar dissolves, about 2 minutes. (If the rum catches on fire while the liquid is reducing, cover with a lid briefly to extinguish the flame.) Add the raisins and cook until the raisins are soft and most of the liquid is reduced, about 5 minutes. In a large bowl, whisk together the egg yolks, ¼ cup of the sugar, and the salt. Combine the milk, cream, the remaining ¼ cup sugar, and the vanilla bean and seeds in a heavy saucepan. Cook over medium heat, stirring occasionally until almost simmering. Slowly pour the milk-and-cream mixture into the egg-and sugar mixture, whisking as you pour.
Return the egg mixture to the saucepan. Cook over medium-low heat, stirring constantly with a heat-resistant plastic or wooden spatula, until the custard reaches 175°F and lightly coats the spatula.
Strain the custard into a clean bowl, discarding the vanilla bean, and cool over an ice bath until room temperature. Put the ¼ cup rum in a small saucepan. Cook over medium heat until reduced by half. (If the rum catches on fire while the liquid is reducing, briefly cover the pan with a lid to extinguish the flame.) Cool to room temperature. Add the reduced rum to the custard and refrigerate for at least 4 hours or up to overnight. Churn in an ice cream machine according to the manufacturer’s instructions. Freeze until scoopable, about 4 hours, depending on your freezer.
To Make the Shortcakes
Preheat the oven to 350°F. Line 2 baking sheets with parchment paper. Sift together the flour, sugar, and baking powder into a large bowl or the bowl of a stand mixer. Stir in the salt. Add the butter and, using 2 dinner knives, a pastry blender, or the paddle attachment of the electric mixer on low speed, mix until the butter is pea-sized. In a small bowl, lightly whisk together one of the eggs and the cream. Add it to the dough and mix until the dough comes together. Gently form the dough into a 5-inch disk. Place the dough on a lightly floured work surface and sprinkle the top lightly with flour. Pat the dough 1¼ inches thick. With a 2-inch-diameter biscuit cutter or drinking glass, cut the dough into rounds. Repeat until you have 10 circles. Place the rounds 2 inches apart on the prepared pans. With a fork, lightly beat the remaining egg in a small bowl. Brush the tops of the shortcakes with some of the beaten egg. Bake until lightly golden, about 20 minutes. Let cool to room temperature.
The shortcakes are best baked and eaten the same day. The sandwiches should be assembled just before serving.