Preheat oven to 325 degrees, with racks in upper and lower thirds. In a small bowl, whisk together flour and salt. In a medium bowl, using an electric mixer, beat butter and granulated sugar on high, scraping down bowl as needed, until light and fluffy, 4 minutes. Add vanilla and beat to combine. Add egg and beat to combine. With mixer on low, gradually add flour mixture and beat just until combined. Transfer dough to a quart-size zip-top bag; with scissors, snip a 3/4-inch hole in one corner (or use a pastry bag with a plain tip). Pipe 54 cookies (3/4 by 2 inches), 1/2 inch apart, onto two parchment-lined baking sheets. Chill until firm, 20 minutes.
Bake until cookies are light golden at edges, about 15 minutes, rotating sheets halfway through. Transfer immediately to wire racks and let cool.
Place chocolate in a medium heatproof bowl set over (not in) a saucepan of simmering water. Stir until melted, 2 minutes. Remove bowl from pan. (Alternatively, melt chocolate in microwave.) In another medium bowl, whisk together confectioners' sugar, peppermint extract, and 4 teaspoons water. With a small offset spatula or table knife, spread chocolate on flat side of half the cookies and mint glaze on the other half. Sandwich cookie halves together.
Piping the cookies: Twist and squeeze the bag so that the batter is in one corner. Snip an opening, then squeeze out excess air. To pipe, hold bag at an angle to sheet with one hand and apply pressure from top with the other.