Preheat oven to 350 degrees. Coat a 9-inch by 5-inch loaf pan with nonstick cooking spray, then line the bottom of the pan with parchment paper.
In a large bowl, whisk together the cake flour, baking powder, and salt. Set aside.
In a food processor, combine the sugar, eggs, and vanilla. Process until combined, about 10 seconds. With the machine running, pour the melted butter through the feed tube in a steady stream. Continue to process until the butter is fully incorporated into the batter.
Make a well in the dry ingredients and slowly pour in the wet ingredients, whisking constantly until most of the lumps are gone. Pour the batter into the prepared loaf pan, wiping up any stray drops of batter.
Bake until a toothpick inserted comes out clean with a few crumbs attached, about 50 to 60 minutes. Rotate the pan halfway through baking.
Cool the cake in the pan for 10 minutes. Remove from pan and peel the parchment from the bottom of the cake. Cool completely.