Start by making a basic pie dough which will serve as the cookie part of the sandwiches. Combine flour, cold butter, and salt in a bowl or a food processor. In the processor, pulse until it resembles sand. By hand, crumble the butter into the flour mixture until it resembles sand.
Add ice cold water a tablespoon at a time. In the processor, pulse a few times until the dough starts to come together. By hand, add your water a bit at a time and bring the dough together, being careful not to knead. If you need to add a bit more water, you can! Once the dough has come together, wrap in plastic wrap and let chill in the fridge for at least 30 mins.
Preheat your oven to 390 F (200 C). Once your dough has rested, roll out your dough until it is about 1/8 inch thick on a floured surface.
Cut out rounds (ours were about 2 inches in diameter). Repeat with any remaining dough.
Once you have your cookie rounds, brush with egg wash.
Prep a piece of parchment paper by sprinkling granulated sugar in an even layer on the top. Place your cookies egg wash side down onto the sugar.
Roll across your cookies with a rolling pin to make them an oval shape.
Once your cookies have been rolled into an oval shape, place on a prepared baking sheet - sugar side up! - and bake for 8-10 minutes until lightly golden. The top should be nice and crispy and caramelized with the sugar layer!
Once your cookies are done, let them cool. In the meantime, make your buttercream by whipping your room temperature butter until light and fluffy. Once it has considerably lightened in color, add your vanilla extract, a pinch of salt, and powdered sugar a bit at a time. Keep mixing until you have incorporated the sugar fully and the mixture looks light and fluffly.
When your cookies are fully cooled, begin assembly by adding buttercream to one half and adding a second cookie to the top. Enjoy!