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, using an electric mixer, beat 1 cup softened butter and granulated sugar on medium-high speed until light and fluffy, about 4 to 5 minutes.
Add eggs one at a time, beating well after each addition. Mix in 1 tablespoon vanilla extract.
With the mixer on low speed, alternately add the flour mixture and milk in three additions, beginning and ending with the flour mixture. Mix just until combined; do not overmix.
Divide the batter evenly between the prepared pans and smooth the tops with a spatula.
Bake for 28 to 32 minutes, or until a toothpick inserted into the center comes out clean.
Allow cakes to cool in the pans for 10 minutes, then turn out onto wire racks to cool completely.
To make the vanilla buttercream filling, beat 1 1/2 cups softened butter on medium speed until creamy, about 2 minutes.
Gradually add powdered sugar, one cup at a time, beating on low speed until combined.
Add 2 teaspoons vanilla extract and 2 tablespoons heavy cream or milk. Beat on high speed for 3 minutes until light and fluffy. Add more cream or milk if needed to reach spreading consistency.
Place one cake layer on a serving plate. Spread about 1 cup of the buttercream evenly over the top.
Place the second cake layer on top and press gently to secure.
Use the remaining buttercream to frost the top and sides of the cake as desired.