In the bowl of a stand mixer fitted with the paddle attachment, beat ¾ cup (170 grams) unsalted butter and ½ cup (60 grams) confectioners' sugar at high speed until creamy, 2 to 3 minutes, stopping to scrape sides of bowl. Beat in 2 teaspoons (8 grams) vanilla extract.
In a medium bowl, whisk together 2 cups (250 grams) all-purpose flour, 1 teaspoon (2 grams) ground cinnamon, 1 teaspoon (2 grams) ground ginger, ½ teaspoon (1 gram) ground cardamom, ¼ teaspoon kosher salt, ¼ teaspoon ground allspice, and 1/8 teaspoon ground black pepper. With mixer on low speed, gradually add flour mixture to butter mixture, beating until combined.
Turn out dough onto a work surface, and roll into a 2-inch thick log (about 5¼ inches long). Wrap tightly in plastic wrap, and refrigerate for at least 2 hours.
Preheat oven to 350°F (180°C). Line baking sheets with parchment paper. Cut log into ¼-inch-thick slices, and place 1 inch apart on prepared pans.
Bake until edges are lightly browned, 8 to 10 minutes. Let cool on pans for 2 minutes. Remove from pans, and let cool completely on wire racks.
Spoon Vanilla Bean Frosting into a pastry bag fitted with a small petal piping tip (Ateco#120). Pipe frosting in a zigzag pattern onto each cookie. Garnish with ground cinnamon, if desired.
In the bowl of a stand mixer fitted with the paddle attachment, beat 1 cup (227 grams) unsalted butter and ¼ teaspoon kosher salt at medium speed until smooth and creamy. With mixer on low speed, add 3 cups (360 grams) confectioners' sugar alternately with 2 tablespoons (30 grams) heavy whipping cream, beginning and ending with sugar, beating until well combined after each addition and stopping to scrape sides of bowl. Beat in 2 teaspoons (12 grams) vanilla bean paste. Increase mixer speed to high, and beat until pale and fluffy, 1 to 2 minutes. Use immediately.
