Preheat the oven to 350ºF. Grease, flour and line with parchment three 6-inch cake pans (Amazon).
Sift together the flour, baking powder, and salt into a medium bowl and set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream together the butter and the sugar for 3 minutes until it's creamy.
Add the eggs, one at a time, scraping down the bowl as needed. Beat well between each addition. After all the eggs are added, add the vanilla and beat for an additionnal 2 to 3 minutes to lighten it. Scrape down the sides of the bowl.
Drop in one third of the flour, then mix on low, followed by half the milk, again mixing it in on low. Scrape down the sides of the bowl.
Add in half the remaining flour, mixing it into the batter on low, the remaining milk, then turn off the mixer and remove it from the stand. Finish adding the last of the flour with the spatula by hand, being sure not to overmix. Divide the batter between the 3 pans, smooth the surface and bang them on the counter to push out any air pockets. Bake the cakes on the middle rack for about 25 to 30 minutes, until the edges are golden brown and begin to pull away from the pan. Use a cake tester to ensure the centre is baked through.
Cool 15 minutes, then unmold on a rack to cool completely.