In a standing electric mixer fitted with the paddle, beat the butter until creamy. Add the brown sugar and beat at medium-high speed until fluffy, 3 minutes. Beat in 1 teaspoon of the vanilla and the 1 whole egg. Add the 2 1/4 cups of flour and 1/2 teaspoon of salt and beat until a soft dough forms. Divide the dough into 2 mounds and transfer to 2 sheets of plastic wrap. Pat each mound of dough into a 1/3-inch-thick square, wrap and refrigerate until firm, 20 minutes.
Preheat the oven to 350° and line 2 large baking sheets with parchment paper. Working with 1 square at a time, on a floured work surface, roll the dough out to a 13-inch square, 1/8 inch thick. Using a 2 1/2-inch biscuit cutter, stamp out rounds and arrange them 1/2 inch apart on the baking sheets; repeat with the second square of dough. Gather the scraps and refrigerate until firm. Reroll and cut out more rounds, using the scraps only once. Refrigerate the cookies just until firm, about 10 minutes.
Bake the cookies in the upper and middle thirds of the oven for about 15 minutes, until lightly browned, shifting the pans halfway through baking. Let cool slightly, then carefully invert the cookies onto racks and let cool completely.
Spread the melted chocolate on half of the cookies and the peanut butter on the other half. Allow the chocolate to set.
In a small saucepan, boil the corn syrup over high heat without stirring until it registers 230° to 235° (thread ball) on a candy thermometer. Meanwhile, in a clean standing mixer fitted with the whisk, beat the 2 egg whites with the cream of tartar and a pinch of salt until firm peaks form. Slowly drizzle in the hot corn syrup and beat at high speed until glossy, about 2 minutes. At medium-low speed, beat in the remaining 2 teaspoons of vanilla and the 1 cup of confectioners' sugar.
Transfer the marshmallow to a pastry bag with a plain 1/2-inch tip or a large zippered plastic bag with the corner snipped off. Pipe 1 1/2-tablespoon mounds of the marshmallow onto the chocolate-covered cookies. Top with the peanut butter–covered cookies and press lightly to spread the marshmallow to the edges. Let the moon pies stand in an airtight container for 2 hours (they will soften slightly) before dusting with confectioners' sugar, cocoa and cinnamon and serving.