Position the rack in the lower third of the oven and preheat to 350° F.
Line the bottoms of two 9 x 2-inch round cake pans with parchment or wax paper.
Sift the flour, baking powder, and salt onto a piece of wax paper; set aside.
In the bowl of a heavy-duty mixer fitted with the paddle blade, beat the butter on medium speed until it is lighter in color; about 45 seconds.
Add the sugar in a steady stream, then stop the machine and scrape down the bowl.
Resume beating, stopping occasionally to scrape down the sides of the bowl, until the mixture is very light in color and texture, 4 to 5 minutes.
Gradually pour in the eggs, about a tablespoon at a time.
(If the mixture looks curdled, stop adding the eggs, increase the speed to high, and beat until it looks smooth and shiny. Return the speed to medium, and add the remaining eggs.) Continue beating until the mixture is ivory-colored.
The entire process of adding and beating the eggs should take 3 to 4 minutes.
Reduce the mixture speed to low.
Add the flour mixture in four additions, alternating with three additions of the milk.
After each addition, beat until smooth and scrape down the sides of the bowl.
Beat in the vanilla with the final addition of milk.
Spread the batter evenly in the prepared pans.
Bake until the tops spring back when the cakes are lightly pressed in the centres and a toothpick inserted in the centres comes out clean, about 25 minutes.
Transfer the layers to wire cooling racks and cool for about 10 minutes.
Invert onto the racks or plates and remove the pans.
Peel off the paper liners and place them back on the layers, sticky sides up.
Invert onto wire racks, right side up, and cool completely on the wax paper.
(The cake can be baked 1 day ahead, cooled, and stored, wrapped tightly in plastic wrap. Or freeze, overwrapped with foil, for up to 2 weeks.) NOTES: it is very important to cream the butter and sugar thoroughly.
This incorporates air and is the key to producing a high-rising cake.
Be sure that all of the ingredients are at room temperature to reduce the chances of curdling.
Scrape down the sides of the bowl often during the creaming and mixing stages.