1 1/2 sticks salted butter (3/4 cup), softened, plus additional for greasing
1 cup sugar, plus 2 tbsps.
1 whole large egg, lightly beaten
1/2 tsp. vanilla
2 cups all-purpose flour
1/2 tsp. cinnamon
1 large egg white, lightly beaten
Stir together butter and 1 cup sugar vigorously with a wooden spoon in a large bowl until combined well, then stir in whole egg and vanilla until smooth. Add flour and stir until just incorporated. Transfer dough to a 24-inch-long log (2 inches wide). Chill, wrapped in wax paper, until firm, at least four hours.
Put oven racks in upper and lower thirds of oven and preheat oven to 375 degrees. Lightly grease two large baking sheets with additional butter. Stir together cinnamon and remaining 2 tablespoons sugar in a small bowl.
Cut chilled log crosswise into 1/8- to 1/4-inch thick slices, arranging slices about one inch apart to fill both baking sheets. Chill remainder of log wrapped in wax paper. Lightly brush sliced cookies with some egg white, then sprinkle with some of the cinnamon-sugar mixture.
Bake cookies, switching position of sheets halfway through baking, until edges are golden, 12 to 15 minutes total. Cool on sheets three minutes, then transfer with a metal spatula to racks to cool completely. Repeat with remaining dough, cooling and greasing baking sheets between batches.
Recipe makes four to six dozen cookies. Dough can be chilled up to five days, or frozen, wrapped in wax paper and foil, for one month. Transfer frozen dough to refrigerator to thaw slightly, at least two hours, before slicing. Cookies can be baked one week ahead and cooled completely, then kept in an airtight container at room temperature.