Preheat the oven to 350 (F). Spray and line two 9-inch cake pans with parchment paper, spraying the parchment paper.
In a medium bowl, whisk together 2 1/2 cups cake flour, 1 tablespoon baking powder, and 1/2 teaspoon kosher salt. Set aside.
In the bowl of a freestanding electric mixer fitted with a paddle attachment, combine 1 cup (2 sticks) unsalted butter and 2 cups granulated sugar, creaming together on medium speed for at least 2 minutes, or until light and fluffy.
Once the mixture is light and fluffy, lower the mixer to its lowest speed. Add 4 large eggs, one at a time, only adding the next egg after the previous egg has been fully incorporated into the batter. Once all the eggs have been incorporated, add 1 tablespoon vanilla and continue mixing until just combined.
Continue to keep the mixer at its lowest speed and add the dry ingredients (from the 2nd step) in three additions, alternating with 1 cup sour cream in two additions, beginning and ending with the dry ingredients. After each addition, beat until just combined — DO NOT OVERMIX at this stage, or you'll end up with a dense, tough cake that will make me sad.
Once the last of the dry ingredients have just been incorporated into the batter, turn off your mixer. Use a rubber spatula to fold in 1/2 cup pistachio paste, making sure to not overmix the batter. Once the pistachio paste has been fully incorporated and the batter has an even, slightly grainy texture, stir in 1/2 teaspoon green food coloring until just combined. There will be a lot of batter at this point, so don't worry if it takes some time to fully incorporate the food coloring. Once the batter is an even green color throughout, divide the batter evenly betwen the two pans and spread the batter out evenly with a rubber spatula. Gently tap each filled pan on the counter 4 or 5 times to remove any air bubbles in the batter. Bake the cakes in the preheated oven for 35 to 45 minutes, or until the top is nicely browned and a skewer inserted into the center of the cake comes out clean. When poked with your finger, the top of the cakes should gently spring back. Transfer to a wire rack and allow to cool completely before frosting and assembling.