In a medium bowl, whisk together the flour, baking powder, and salt.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy.
Beat in the egg and vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Cover the dough and refrigerate for at least 30 minutes.
Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper.
Roll out the chilled dough on a lightly floured surface to about 1/8-inch thickness.
Using a cookie cutter (or a knife), cut out oval shapes. Place them about 1 inch apart on the prepared baking sheets.
Bake for 10-12 minutes, or until the edges are lightly golden.
Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.
For the filling: Melt the semisweet chocolate and butter in a heatproof bowl set over a saucepan of simmering water (double boiler), or in the microwave in 30-second intervals, stirring until smooth.
Once the cookies are completely cool, spread a thin layer of the chocolate filling on the flat side of one cookie, then top with another cookie to form a sandwich. Repeat with remaining cookies.
Let the filled cookies sit at room temperature for the chocolate to set.
Notes
Store Milan cookies in an airtight container at room temperature for up to 3 days.