Preheat the oven to 350 degrees F (175 degrees C). Spray three 8-inch round cake pans with cooking spray. Line each pan with a round piece of parchment paper, and spray with cooking spray.

Pulse pistachios in a food processor until finely ground, about 10 to 15 pulses. Transfer 1 1/2 cups ground pistachios to a bowl with flour, baking powder, salt, and baking soda; whisk until well combined and set aside. Reserve remaining ground pistachios for finishing the cake.

Beat butter and white sugar in a stand mixer fitted with a paddle attachment on medium-high speed until light and fluffy, about 3 minutes. Add egg whites and beat on high speed until combined, about 2 minutes.

Mix in sour cream, vanilla extract, and almond extract until combined, about 1 minute. Add flour mixture in batches on low speed until completely incorporated, about 2 minutes. Mix in milk and food coloring until just combined, about 1 minute.

Divide batter evenly among the prepared cake pans. Gently tap each pan on the countertop to spread and flatten batter.

Bake in the preheated oven until a toothpick inserted into the center of each pan removes cleanly, 22 to 24 minutes. Cool cakes in the pans for about 10 minutes. Run a table knife around the edges to loosen cakes, then invert carefully onto a serving plate or cooling rack. Let cool completely, about 30 minutes.

While the cakes are cooling, prepare buttercream: beat cream cheese and butter in a stand mixer fitted with a paddle attachment on medium-high speed until light and fluffy, about 3 minutes.

Mix in powdered sugar in 1/2-cup increments on medium-low speed, allowing sugar to be completely incorporated before adding more, until combined, about 4 minutes. Mix in vanilla and food coloring on medium-high speed until light and fluffy, about 2 minutes.

Place one cake layer on a serving plate or cake stand. Top with 1 cup buttercream and 1 tablespoon ground pistachios, and spread evenly over the surface.

Place second cake layer on top; top with 1 cup buttercream and 1 tablespoon ground pistachios, and spread evenly over the surface.

Place third cake layer on top, and spread 1 cup buttercream evenly over the surface.

Spread 2 cups buttercream evenly around sides of assembled cake and smooth with a spatula. Transfer cake to the refrigerator and chill for 30 minutes.

Remove cake from the refrigerator and spread remaining buttercream over top and sides of cake until sides are smooth and top is billowy. Using a small offset spatula, make a spiral pattern on top of the cake. Press remaining ground pistachios into sides of the cake.

Slice and serve immediately or chill in an airtight container until ready to serve. If chilled, allow cake to come to room temperature before serving.