Preheat the oven to 400˚ F. Make the shortcakes: Pulse the flour, granulated sugar, baking powder, allspice, ginger, nutmeg and salt in a food processor. Pulse in the butter until it is in pea-size pieces, then pulse in the heavy cream and egg until the dough comes together.
Bake the shortcakes: Drop 6 scoops of dough onto a baking sheet, brush with heavy cream and sprinkle with coarse sugar. Bake until golden, 20 to 25 minutes. Let cool, then split in half.
Make the filling: Toss the peaches and mango with the granulated sugar and lime juice. Beat the heavy cream with the confectioners' sugar until stiff peaks form; fold in the rum and vanilla. Fill the shortcakes with the fruit and whipped cream.