To make the cookies: To the bowl of a stand-up mixer, add the butter, sugar, powdered sugar and salt. Beat until light and fluffy, about 3 minutes. Add the egg, egg yolk, vanilla extract and beat once more until combined. In two batches, add the flour, being sure each addition is combined before adding the next one. Divide the dough in half, transfer it to two sheets of plastic wrap (it’ll be quite soft!), shape into a disc and transfer to the fridge. Allow to chill for about 1 hour. Ideally it should be at least 3 hours.
Turn the oven up to 375 degrees F. Line two baking sheets with parchment paper and set aside.
Remove the first disc of dough from the fridge. Allow to come to room temperature for about 10 minutes. Liberally flour a rolling pin and work surface. Roll out the cookie dough to a 1/8-inch thickness. Stamp out the cookies with a cookie cutter of your choice (I used a round cookie cutter that I shaped into an oval to replicate the traditional milano cookie shape). Transfer the cookies to the baking sheet, spacing them about 1-inch apart (these spread slightly). Re-roll the scraps to get a few more cookies. Repeat with the second disc of dough.
Bake the cookies, one baking sheet at a time, for about 8 to 10 minutes, until lightly golden brown around the edges. Cool on the baking sheet before transferring to a cooling rack. Repeat with the remaining cookies.
To make the chocolate: Add the chocolate and heavy cream to a very dry stainless steel or glass bowl. Fill a saucepan with a few inches of water and set it over medium heat. Nestle the bowl with the chocolate atop the saucepan and melt the chocolate slowly, about 3 minutes. If you own a microwave, you can simply add the glass bowl with the chocolate and heavy cream to the microwave for about 30 seconds until melted. When the chocolate is melted, stir in the peppermint extract.
Flip over half of the cookies (these will be the cookie bottoms). Drop about a teaspoon of melted chocolate onto each of the cookies bottoms and spread it out just a bit. Top with the “tops” of the cookies and sandwich the two together. Repeat until you’ve worked your way through all of the cookies.