Mise en place. Preheat oven to 350˚F. Grease and flour two 8-inch round cake pans and set aside. Then, measure out all of your ingredients before you begin.
Grind the pistachios. Add shelled pistachios to a food processor and pulse until they become a fine meal, similar to almond flour. Being careful not to blend for too long to prevent the pistachios from turning to nut butter.
Mix the dry ingredients. In a mixing bowl, whisk the ground pistachios, flour, baking powder, matcha powder, and salt together until evenly combined. Set aside.
Cream the butter and sugar. In a large bowl with an electric mixer, beat the dairy free butter, vegetable shortening, and sugar together on medium speed until light and fluffy, about 2 minutes.
Add the wet ingredients. Beat in the eggs, one and a time, followed by the milk, yogurt, and extracts. Mix until combined.
Combine wet and dry. Add the flour mixture, about 1/2 cup at a time, beating on low until fully combined, scraping the sides of the bowl as needed. The color should turn nice and green when the wet ingredients mix with the matcha.
Bake the cake layers. Divide the batter evenly between your prepared pans. Bake for 35–40 minutes, or until an inserted toothpick comes out clean and the tops spring back when gently pressed. Remove the cakes from the oven and allow them to cool to room temperature before frosting.
Melt the white chocolate. Add white chocolate chips to a small microwave-safe bowl. Microwave for two 30-second intervals, stirring between each, until fully melted and smooth. If needed, continue to heat for 10 second intervals, until the chocolate is melted.
Make the frosting. In a large bowl with an electric mixer, beat the shortening or dairy free butter with salt, vanilla extract, and dairy free milk until combined. Add the melted chocolate and about 1 cup of powdered sugar, making until incorporated. Continue adding powdered sugar, 1 cup at a time, until you reach your desired consistency. Frost cake as desired.