1. Preheat oven to 350 degrees, with racks in upper and lower positions. On generously floured pieces of parchment paper, roll out dough to about 1/16 inch thick. Brush off excess flour. Slide dough and parchment onto baking sheets, and freeze 10 minutes.

2. Using a 2 3/8-inch fluted cutter, cut out rounds, rerolling scraps to get a total of 72. Transfer to parchment-lined baking sheets, and freeze 10 minutes.

3. Bake cookies 5 minutes. Remove sheets from oven; tap pans firmly on counter to flatten cookies. Return to oven, rotating sheets from top to bottom and front to back; bake until cookies are set and edges have darkened slightly, about 5 minutes more. Transfer sheets to wire racks to cool completely.

4. With an electric mixer on medium speed, beat cream, confectioners' sugar, and vanilla just until stiff peaks form. Arrange 7 cookies in a circle on a cake stand or a serving plate, sides touching; place 1 cookie in center. Spread 3/4 cup whipped cream over cookies, leaving a 3/4-inch cookie border. Repeat, making 8 more layers each of cookies and whipped cream. Chill at least 1 hour.

5. Just before serving, spoon some of the oranges and their syrup over top. Serve remaining oranges on the side.