Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans and line the bottoms with parchment paper.
In a medium bowl, whisk together the sifted flour, baking powder, and salt. Set aside.
In a large bowl, beat the 1 cup softened butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 4-5 minutes.
Add the eggs one at a time, beating well after each addition. Mix in the 1 tablespoon vanilla extract.
With the mixer on low speed, alternately add the dry ingredients and the milk to the butter mixture, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Divide the batter evenly between the prepared pans and smooth the tops with a spatula.
Bake for 28-32 minutes or until a toothpick inserted into the center comes out clean.
Remove the cakes from the oven and allow them to cool in the pans for 10 minutes. Then run a knife around the edges, invert the cakes onto wire racks, peel off the parchment, and let cool completely.
To make the vanilla buttercream filling, beat the 1 1/2 cups softened butter on medium speed until creamy, about 2 minutes.
Gradually add the powdered sugar, 1 cup at a time, beating on low speed until incorporated. Add the heavy cream, vanilla extract, and salt, then beat on high speed for 3 minutes until light and fluffy.
Place one cake layer on a serving plate. Spread about half of the buttercream evenly over the top.
Place the second cake layer on top and spread the remaining buttercream over the top and sides of the cake as desired.
Refrigerate the cake for at least 30 minutes before serving to set the filling and frosting.