Make the cookie dough: In the bowl of an electric mixer fitted with the paddle attachment, cream butter, 1/2 cup plus 2 tablespoons sugar, the vanilla, and salt together until smooth. Add 1 egg and egg yolk, beating to combine, scraping the sides of the bowl with a rubber spatula. Add flour gradually, and continue beating until just blended. Transfer the dough to a sheet of plastic wrap, and shape into a rectangle. Wrap the dough well. Chill in the refrigerator until firm, about 2 hours.
Make the filling: In a shallow, medium saucepan, combine lemon rind, cinnamon stick, figlets, cognac, 1 cup sugar, and 2 cups water over medium-high heat. Bring to a boil, and reduce heat to low; simmer until the liquid has evaporated, about 30 minutes, stirring occasionally. Remove lemon rind and cinnamon stick, and let cool.
Place fig mixture in the bowl of a food processor, and pulse until smooth, 20 to 30 seconds.
Remove the cookie dough from the refrigerator, and divide in half. On a parchment-lined work surface that has been lightly floured, roll one half of the dough into a rectangle about 1/8 inch thick. Trim the dough to 9 by 12 inches with a pizza wheel or a knife. Cut the rectangle lengthwise into three strips, 3 by 12 inches. If the dough becomes soft, it may be necessary to chill it again.
Transfer the filling to a pastry bag fitted with a #8 or #808 plain tip. Pipe the filling down the center of each strip. Brush beaten egg along one of the 12-inch sides of dough. Fold dough over fig filling to enclose, pressing gently to seal. Transfer to a baking sheet lined with a parchment paper, seam side down. Repeat with the remaining dough and filling. Chill in refrigerator for 30 minutes.
Preheat oven to 350 degrees. Bake cookies until light golden, 20 to 25 minutes. Remove to a cooling rack. When completely cooled, transfer to a cutting board, and cut each log into eight 1 1/2-inch cookies with a serrated knife.