Add the softened butter and powdered sugar to the bowl of a stand mixer fitted with the paddle attachment. Cream the butter and sugar until smooth and fluffy, about 2 minutes.
Add in the egg, vanilla, and lemon zest and mix until fully incorporated.
Add in the flour, baking powder, and salt and mix until just combined. The dough should come together into a ball but will feel dry. If the dough doesn't come together, you can add a teaspoon of milk or water at a time until it does.
Press the dough into a disc and wrap tightly in plastic wrap. Refrigerate until firm (1-2 hours minimum but up to 24 hours).
Line a baking sheet with parchment paper.
On a lightly floured surface, roll out the dough ¼ inch thick and cut into circles. Re-roll and cutout the scraps.
Place cutout cookies 1-2 inches apart on your baking sheet (they won’t spread). Place the baking sheet in the freezer for a minimum of 20 minutes (or the fridge for 40 minutes) before baking to allow the cookies to firm back up. This ensures the cookies won't spread during baking.
While the cookies are chilling, preheat your oven to 350°F.
Bake for 8-10 minutes or until the edges are set. Bake on the middle rack of the oven and rotate the pan halfway through baking for the best results. It usually takes 2-3 batches to bake all the cookies.
Let the cookies cool on the baking sheet for 2-3 minutes, then remove and let them finish cooling on a cooling rack. Let the cookies cool completely before assembling.
In a stand mixer fitted with the paddle attachment, cream together the softened butter and powdered sugar until smooth and creamy.
Add in the raspberry puree and lemon juice. Beat the buttercream for 3-4 minutes on high speed until light and fluffy.
Transfer the buttercream to a piping bag fitted with a large round tip. Pipe a small amount of filling into the center of a cookie and top with a second cookie. Dust with powdered sugar if desired.
To make raspberry puree, place 5-7 whole fresh raspberries in a blender or food processor and blend until smooth. If desired, you can strain any chunks or seeds.
Store in an airtight container at room temperature for up to 5 days.
You can also freeze the cookie dough for up to 3 months. If frozen, allow the dough to thaw overnight in the refrigerator before rolling baking.