Position a rack in the center of the oven and heat the oven to 350°F. Coat the bottoms and sides of three 9-by-2-inch round cake pans with nonstick baking spray or butter and line with parchment circles.
Sift together the flour, baking powder, and salt into a medium bowl.
In the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl, using a handheld mixer), cream the butter, corn syrup, and sugar on medium speed until light and fluffy, about 4 minutes.
Whisk together the eggs, crème fraîche, and vanilla in a small bowl, then pour the mixture into the creamed butter and beat until smooth. Scrape the bottom and sides of the bowl and the paddle and mix for 30 seconds. With the mixer on low speed, add the dry ingredients in two batches, mixing for 30 seconds after each addition. Scrape the bowl again and mix for 15 seconds.
Divide the batter among the prepared cake pans, spreading it evenly. Bake for 15 minutes. Rotate the cakes and bake for an additional 10 to 12 minutes, until the cakes appear firm and have a matte finish and a toothpick inserted in the center comes out clean. Let the cakes cool completely in the pans on cooling racks.
Wrap the cakes in plastic wrap and refrigerate until you are ready to assemble the cake.