In the bowl of a stand mixer fitted with a paddle attachment, cream the softened butter until light and fluffy, about 2-3 minutes. Add in the vanilla extract and heavy cream, mix to combine.
With the mixer on low add in 1/2 a cup of powdered sugar at a time, mixing until combined each time. Scrape down the sides occasionally. Continue to add powdered sugar until you have reached a desired consistency, you may not need it all. Mix in the lemon zest and zest.
If your frosting is too thick, add more cream, if it is too thin, add more powdered sugar, about 1 tablespoon at a time. Spread or pipe the frosting onto the cooled cookies.
For the Shortbread Cookies:
Preheat oven to 350°F. Line baking sheets with parchment paper and set aside.
In a small mixing bowl combine the flour, baking powder, baking soda, cream of tartar, and salt. Stir to combine, set aside
In the bowl of a stand mixer fitted with a paddle attachment, cream the butter and sugar on medium speed until fluffy, 3-4 minutes. Add in the vanilla and the egg. Scrape down the sides. Mix in the lemon juice and zest.
Add in the dry ingredients and mix on low until combined. Gently mix in the poppy seeds.
Transfer the dough out onto a clean, lightly floured surface. Roll out to 1/2? thick. Using a cookie cutter, cut the dough into shapes. Re-roll the remaining dough and continue cutting until all dough is used.*
Arrange the cookies on the prepared baking sheets, about 2? apart. Chill for 30 minutes. Once chilled, remove from the fridge and bake at 350 for 4-6 minutes.* Allow to cool for 4-5 minutes on the baking sheets before transferring to a cooling rack.
Once the cookies have cooled completely, pipe or spread on the buttercream, sandwiching together. Store in an airtight container in the fridge for 3-4 days.
Notes
*It doesn’t seem like a lot of dough, but you get a lot of cookies from the dough scraps you re-roll, you can also scoop this dough with a portion scoop. *Baking time is dependent on the size of the cookies, I would start with 4-6 minutes and increase baking time as needed.