Preheat oven to 200 degrees. Invert a 7-cup bombe mold (see recipe, above) onto 1 parchment-lined baking sheet, and trace perimeter. Turn over parchment. Brush traced circle with butter, and dust with flour. Brush another parchment-lined baking sheet all over with butter, and dust with flour.
Beat together almonds, 1/2 cup superfine sugar, and the confectioners' sugar in a large bowl with a mixer. Beat together egg whites with a mixer until soft peaks form. Beat in remaining 1/2 cup superfine sugar, 1 tablespoon at a time, beating until stiff, glossy peaks form. Fold into nut mixture.
Spread meringue into traced circle to form a round slightly thicker than 1/4 inch. Fit a large pastry bag with a 1-inch flat tip with a serrated edge (such as Ateco #9897). On the second baking sheet, use serrated side to pipe 3 overlapping lines to create a 3-inch square. Repeat to create 11 more squares.
Bake dacquoise round and cookies until they are dry on top and lift easily from parchment, about 1 hour, rotating halfway through. Turn off heat; let cool in oven 1 hour. If it isn't too humid, dacquoise can be stored at room temperature, well wrapped, 1 day (if humid, use immediately).