For the vanilla cake: Preheat the oven to 350 degrees F. Coat three 6-inch-round cake pans with cooking spray and line the bottoms with parchment paper; spray the parchment and dust the pans with flour, tapping out the excess.
Whisk the baking powder, salt and 3 cups flour together in a bowl until combined. Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce the speed to medium, then beat in the eggs, one at a time, scraping down the bowl as needed. Beat in the vanilla. (The mixture may look separated at this point.) Beat in the flour mixture in 3 batches, alternating with the milk and ending with the flour, until just smooth.
Divide the batter among the prepared pans. Bake until the cakes are lightly golden on top and a toothpick inserted in the middles comes out clean, 30 to 35 minutes. Transfer to racks and let cool 10 minutes, then run a knife around the edges of the pans and turn the cakes out onto the racks to cool completely. Remove the parchment.
For the buttercream: Mix together the confectioners' sugar and butter in a stand mixer fitted with the whisk attachment. Mix on low speed until well blended, then increase speed to medium and beat for another 3 minutes.
Add vanilla and cream and continue to beat 1 minute more, adding more cream if needed for spreading consistency.
Trim the tops of the cakes with a long serrated knife to make them level, if desired. Spread frosting on each layer, stack layers and cover with frosting.