1. Preheat oven to 350 degrees F. Line two baking sheets with parchment paper or silicone baking mat; set aside.
2. In a medium bowl, whisk flour and baking powder; set aside.
3. In a large bowl, cream butter and sugar until pale and fluffy. Beat in the egg and vanilla extract. On low speed, beat in the flour mixture just until combined.
4. On a lightly floured surface, roll out the dough to a little less than ¼-inch thick. Using a 1¾-inch round or fluted cookie cutter, cut out rounds. Place about one inch apart on prepared baking sheets. Bake for 8-10 minutes, just until the edges start to color. Watch closely, as they can get too dark rather quickly! Transfer to wire racks to cool completely.
5. Spread bottom side of half of the cookies with about ¼ teaspoon of jam. Top with remaining cookies.
6. To make icing, whisk together the powdered sugar, vanilla extract and milk, adjusting the milk and powdered sugar to reach the desired consistency. Spread over the tops of the cookies. Let stand until icing is completely dry. Cookies can be stored in an airtight container at room temperature.