Make the frangipane:
In a food processor, blend the almonds and sugar until the nuts are finely ground. Add the egg, egg yolks, butter, rum, and almond extract; process until smoothly blended. Cover and refrigerate the filling.
Assemble the pastry
Position a rack in the lower third of the oven; heat the oven to 375°F. Brush the bottom and sides of an 8-inch round, 2-inch deep cake pan with melted butter. Stack the phyllo sheets and halve them crosswise with a sharp knife to make 12x81/2-inch sheets. Keep them covered with plastic and then a damp towel. Brush one half sheet with the melted butter and ease it into the prepared pan, allowing the ends to extend up the sides of the pan. Sprinkle with about 2 tsp. of the chopped almonds. Brush another half sheet with butter and lay it over the first sheet perpendicular to the first sheet (so it forms a cross). Sprinkle with another 2 tsp. chopped almonds. Continue buttering, layering, and sprinkling, crisscrossing the sheets so they evenly cover the bottom and sides of the pan, using another nine half sheets of phyllo. Spoon the frangipane into the phyllo-lined pan, spreading evenly. Snip the edges of the phyllo to leave an even 1-1/2-inch border. Fold the border over the filling; brush the top of the phyllo with melted butter.
Using the cake pan as a guide, cut six 8-inch rounds out of six more phyllo half-sheets. Brushing each layer generously with melted butter, stack the rounds on plastic wrap. With a sharp paring knife, cut a 3/4-inch circle in the center to allow steam to escape during baking. Score the top layers, starting 1/2 inch from the circle vent, with 8 semicircle cuts, pinwheel-style. Transfer the stack to the top of the filling, patting gently to seal (or do the scoring after you transfer the top).
Bake until puffed and golden, 30 to 40 minutes. Remove the Pithiviers from the oven and let cool for about 10 minutes before carefully removing it from the pan and transferring it, vent side up, to a serving platter. Serve warm or at room temperature.