Place one oven rack in the top third of the oven and one in the bottom third. Preheat the oven to 350°F.

In the bowl of a stand mixer fitted with a paddle attachment, combine the butter, powdered and granulated sugar. Beat on low speed for 30 seconds, just until combined. Increase the speed to medium and beat for about 3 minutes, until light and fluffy.

Scrape down the sides and bottom of the bowl. Add the egg and vanilla extract. Beat on medium-low speed for about 1 minute. (The egg may not be fully incorporated, but the dough will smooth out once the flour is added.)

Add the salt and then the flour in two parts, alternating with the milk. Mix on low speed between each addition, just until combined. Scrape down the sides and bottom of the bowl as needed.

Transfer the dough to a piping bag. Cut a ¾-inch wide opening from the tip of the piping bag.

Line two large baking sheets with parchment paper. Pipe log-shaped cookies about 2½-inches long and about ¾ to 1 inch wide, leaving about 2 inches of space between each.

Bake the cookies for 9 to 10 minutes, rotating the trays from top to bottom rack and giving them each a 180-degree turn after 5 minutes of baking, until the tops are dry and the bottom edges are lightly golden.

Turn off the oven and crack the oven door open about 6 inches. Leave the cookies in the oven for 10 more minutes to help dry them out and crisp. Transfer to a cooling rack to cool completely.

Match cookies that are similar in size and shape to create pairs.

While the cookies are cooling, place the chocolate chips in a medium, heatproof bowl.

Fill the base of a double-boiler or small pot with 1 inch of water. Place over medium-high heat and bring the water to a simmer. Reduce the heat to medium-low and place the bowl of chocolate chips over the top, ensuring that the bottom of the bowl does not touch the water.

Cook for 4 to 6 minutes, stirring occasionally, until the chocolate is fully melted and smooth. Remove the bowl from the heat.

Let the chocolate cool for about 10 minutes, stirring occasionally, until it thickens slightly.

Spread about 1 teaspoon of the melted chocolate across the bottom of one of the cookies, reaching almost to the edges. Top with its similarly sized cookie and gently press the two together until the chocolate reaches the edges (but does not spill over).

Set aside the cookies to cool completely. Repeat with the remaining pairs of cookies.

Let the chocolate cool and set completely.