Milano Cookies

In the bowl of an electric mixture, use the paddle attachment to beat the butter and granulated sugar on medium-high speed until light and fluffy, about 3 minutes. Scrape down the bowl. Add the vanilla and egg and beat to combine. With the mixer on low, gradually add the flour and salt and beat until just 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 about 24 logs about 3 inches long, spacing 1/2 inch apart, onto two parchment-lined baking sheets. For best results, chill until firm, about 30 minutes.
Preheat the oven to 325°F. Bake until cookies are light golden at the edges, about 15 minutes, rotating sheets halfway through. Transfer to wire racks and let cool.
Melt the chocolate chips in a small heat-proof bowl in the microwave for about 1 1/2 minutes, stirring every 15 seconds, until smooth.
If desired, in a separate bowl, whisk together powdered sugar, peppermint extract, and 3 to 4 teaspoons of water.
With a small offset spatula, spread the melted chocolate on the flat side of half the cookies and mint glaze on the other half (if using). Sandwich cookie halves together.
Cookies can be stored in an airtight container up to 5 days.