To make the frangipane: With an electric mixer, beat together the almond paste, sugar, and butter. Add the rum and egg and continue to mix until smooth.

To assemble the galette: Adjust an oven rack to the center position and heat the oven to 400 degrees F. With a serrated knife, cut the pineapple half into 1/2-inch slices, and then cut the slices into 1-inch pieces.

Cover a baking sheet, preferably one without sides, with kitchen parchment. (If your baking sheet has sides, flip it over and use the back.)

On a lightly floured surface, roll the galette dough into a 15-inch round. Transfer the dough by folding it in half, picking it up, and unfolding it on the lined baking sheet. Smear the frangipane over the dough, leaving a 2-inch border without frangipane. Arrange the pineapple slices in an even layer over the frangipane. Lift the edges of the dough and fold them inward over the filling, pleating as you go, to create a folded-over border. Brush the border with the melted butter and sprinkle the entire galette with the sugar. Bake until the sugary crust has browned and the frangipane is set, 30 to 35 minutes Let cool for about 5 minutes, and then slide the galette off the parchment and onto a cooling rack. Let cool for another 20 minutes before slicing.

In a large bowl, mix together the flour, sugar, and salt. Cut in the chilled butter using a stand mixer, a food processor, or a pastry blender until the butter is evenly distributed but still in large, visible pieces. Add the ice water all at once to the flour and butter. Mix the dough just until it begins to come together (if using a stand mixer or a food processor, be especially careful not to overmix the dough). Gather the dough with your hands--don't worry if you see streaks of butter--and shape it into two disks. Wrap the disks in plastic and refrigerate for at least 1 hour.