For the cookies: Preheat the oven to 350 degrees F and line two rimmed baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda and salt.
Cream the butter and granulated sugar together with an electric mixer. Scrape the bowl and mix in the eggs. Scrape the bowl again and add the sour cream, vanilla and lemon zest. Mix to combine, then add the dry ingredients and mix until just combined—do not overmix.
Using a 3-ounce ice cream scoop, scoop the batter for 6 cookies onto each prepared baking sheet. They will spread, so make sure there is plenty of room around each scoop. Bake, rotating the baking sheets once during baking so they bake evenly, until the edges of the cookies just start to brown, 15 to 18 minutes.
Let the cookies cool for a few minutes, then transfer to a wire rack to finish cooling. If you don't cool on a rack the cookies can steam themselves and get gummy.
For the icing: Whisk together the milk, corn syrup, vanilla and salt in a liquid measuring cup. Add the powdered sugar to the bowl of a stand mixer fitted with the paddle attachment (or use a hand mixer). Add the milk mixture and mix on low speed until combined. Increase the speed to high and whip until fluffy and smooth. Stir in the cardamom and lemon zest.
Transfer 3/4 cup of the icing to a small bowl and whisk in the cocoa powder until well combined. Divide the remaining icing evenly between 2 more small bowls; add the pistachio paste to one of the bowls and whisk well to combine.
To decorate: Spread the pistachio icing on half of the flat side of each cookie. Refrigerate the cookies for 5 minutes, then spread the other half of each cookie with the white cardamom icing. Drizzle the chocolate icing over the tops. Sprinkle with the ground pistachios, if desired.
Let the icing set for 5 minutes in the fridge and enjoy!
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