Preheat oven to 325 degrees, with racks in upper and lower thirds. Prepare baking sheets by lining them with parchment paper.
In a small bowl, whisk together flour and salt.
In a medium bowl, using an electric hand 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.
On parchment lined paper, create small oval doughs and line on parchment paper, leaving a little room for spreading.
Bake until cookies are light golden at edges, about 15 minutes, rotating sheets halfway through. Transfer immediately to wire racks and let cool.
Melt chocolate in a double boiler by adding chocolate in a medium heatproof bowl set over (not in) a saucepan of simmering water. Stir until melted about 2 minutes. Remove bowl from pan and set aside. In another medium bowl, whisk together confectioners' sugar, peppermint extract, and 3- 4 teaspoons water.
Take two cooled cookies. With a small spatula or table knife, spread chocolate on flat side of one cookie and then the mint glaze on the other half. Sandwich cookie halves together.
Enjoy! Store in an airtight container in fridge for up to a week or on counter for up to 3 days.