Preheat oven to 350Â°F with rack in middle. Butter a 9x13 cake pan, then line bottom with a rectangle of parchment paper and butter parchment. Dust with flour, knocking out excess.
Sift together cake flour, baking powder, baking soda, and salt.
Beat together butter, sugar, and vanilla with an electric mixer at high speed until pale and fluffy, 3 to 5 minutes.
Beat in eggs, 1 at a time, at medium speed.
At low speed, mix in flour mixture in 3 batches, alternating with yogurt, beginning and ending with flour mixture, and mixing until just combined.
Spread batter evenly in pan and rap pan on counter several times to eliminate air bubbles.
Bake until cake pulls away from sides of pan and a wooden pick inserted in center comes out clean, 35 to 45 minutes. Cool in pan 10 minutes, then run a knife around edge. Invert onto a rack and discard parchment. Cool completely, about 1 hour.
When cake is cool, stir together confectioners sugar, corn syrup, cream, and vanilla until smooth.
Put cake on a platter and spread icing on top in a thick layer, letting it run down sides. Let icing set at least 15 minutes.