This simple yet festive layer cake gets its smooth and creamy pistachio infusion from a nut paste used to make the pastry cream.
Preheat oven to 325 degrees. Butter two 8-inch round cake pans. Line bottoms with parchment; butter parchment and dust with flour, tapping out excess. Whisk together flour, baking powder, and salt in a medium bowl. In a mixer fitted with the paddle attachment, beat butter with sugar on medium-high speed until light and fluffy, about 5 minutes. Add vanilla and beat until combined. Add eggs, one at a time, and beat until incorporated, scraping down bowl as needed. Reduce speed to low and add flour mixture in three batches, alternating with milk in two batches.
Divide batter evenly among prepared pans. Smooth tops with an offset spatula. Bake until tops are golden brown and a tester inserted into centers comes out clean, 45 to 50 minutes. Let cool in pans 10 minutes, then invert cakes onto a wire rack, remove parchment, reinvert, and let cool completely. With a serrated knife, trim tops of cakes so they're level.
Stir together pastry cream and 1/3 cup buttercream. Spread pastry-cream mixture over one cake layer and top with second cake layer, cut-side down. Freeze until firm, about 30 minutes. Spread about 1 cup of buttercream in a thin layer on top of sides of cake (to prevent crumbs from pulling through final layer of frosting; it's okay if crumb coating is uneven, as it will be covered up later). Refrigerate until firm, about 30 minutes. Frost cake with remaining buttercream and refrigerate until icing is firm, at least 30 minutes and up to overnight.
Use a 2-inch round biscuit cutter to mark an overlapping circle pattern around top edge of cake. Place royal icing in a piping bag fitted with a small round tip (such as Ateco #2). Pipe overlapping circles, using markings as a guide. Use a small damp brush to correct any wobbly piping. Cake can be stored in refrigerator up to 1 day; bring to room temperature 1 hour before serving.