Make the Cake: Preheat oven to 325 degrees. Butter pans; line with a parchment circle cut to size. Butter and flour parchment; set pans aside. Sift together flour and baking powder three times; set aside.
Cream butter and sugar in a mixer bowl until fluffy. Add egg yolks one at a time, beating until incorporated after each. Add flour mixture to egg mixture in three additions, alternating with milk, and starting and ending with flour.
Whisk egg whites with the salt in teh clean bowl of a mixer until stiff but not dry. Mix 1/4 of the whites into the batter; then fold in remaining whites.
Pour batter into pans. Bake, rotating pans halfway through, until tops are golden brown and a tester inserted into centers comes out clean. Cool cakes in pans for 20 minutes. Run a knife around outsides of cakes; turn out onto wire racks. Remove parchment, and reinvert cakes to cool, top side up. If making ahead, wrap cooled cakes in plastic wrap; refrigerate at least overnight before using. The cakes will keep, refrigerated, for 2 days, or for 3 months frozen.
Make the Buttercream: Put the sugar and egg whites in a heatproof mixer bowl, and set over a saucepan of simmering water. Heat, whisking constantly, until the sugar dissolves and the mixture registers 140 degrees on an instant-read thermometer.
Transfer bowl to stand of mixer. Whisk egg-white mixture on medium-high speed until fluffy and cooled, about 10 minutes. Continue beating until stiff, glossy peaks form.
Reduce the speed to medium low, and add the butter by the tablespoon, beating well after each addition. Beat in the vanilla. Buttercream can be refrigerated in an airtight container up to 1 week or frozen up to 1 month. Bring to room temperature, and beat on low speed until smooth, about 3 minutes, before using. If buttercream should separate and look "curdled," simply beat it until smooth again.
Approximate batter amounts and baking times for 2-inch-high pans (use a 3-inch pan for the 8-inch tier): 14-inch tier, 15 cups batter in each of 3 pans (45 cups total), 1 hour 40 minutes; 12-inch, 10 cups each in 2 pans, 1 hour 35 minutes; 10-inch, 7 cups each in 2 pans, 1 hour 15 minutes; 8-inch, 6 1/2 cups in 1 pan, 1 hour 10 minutes; 6-inch, 3 1/2 cups each in 2 pans, 1 hour.
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