2 sticks (1 cup) unsalted butter, cut into chunks, plus more for buttering the pans
16 ounces all-purpose flour (about 3 1/2 cups plus 3 tablespoons)
2 1/2 teaspoons baking powder
22 ounces granulated sugar (about 3 cups plus 2 tablespoons)
3/4 teaspoon salt
1 1/2 cups milk
1 cup egg whites (from about 10 large eggs)
2 tablespoons vegetable oil
1 1/2 tablespoons vanilla bean paste
1/2 cup pecans, chopped
3 tablespoons unsalted butter
One 14-ounce can sweetened condensed milk
1 teaspoon vanilla extract
1 stick (8 tablespoons) unsalted butter, at room temperature
2 cups confectioners' sugar, sifted
1/2 teaspoon vanilla bean paste
2 tablespoons milk
For the cake: Preheat the oven to 350 degrees F. Butter the sides and bottoms of two 8-inch round cake pans. Line the bottoms and sides with parchment if desired (for easy removal).
Add the flour, baking powder, granulated sugar and salt to the bowl of a stand mixer fitted with the paddle attachment and combine well. In a separate bowl, combine the milk, egg whites, vegetable oil and vanilla bean paste and whisk to combine. Set aside.
Turn the mixer to low and add the chunks of butter slowly to the dry mix. Continue to beat on low until there are no chunks of butter remaining and the mixture becomes crumbly. Add a third of the wet ingredients to the bowl and then turn the speed to medium. Mix until a light paste forms, scraping down the sides of the bowl. Add half of the remaining wet ingredients and beat on medium-high speed for 4 minutes. Scrape down the sides and bottom of the bowl.
Add the remaining wet ingredients and beat on medium speed for 5 minutes. Divide the batter evenly between the two prepared cake pans and bake until a skewer comes out clean and the sides have begun to pull away from the sides of the pans, 40 to 50 minutes. Cool in the pans on a rack, 10 minutes, then invert onto the rack to remove the pans and parchment and cool completely.
For the filling: Roast the pecans in the butter in a skillet over medium-high heat. Add the condensed milk and stir constantly until the mixture thickens. Remove from the heat and stir in the vanilla extract and salt. Cool completely before using.
For the buttercream: Beat the butter with a hand mixer, the paddle attachment of a stand mixer or a wooden spoon until smooth and fluffy. Gradually beat in the confectioners' sugar until fluffy. Beat in the vanilla bean paste. Pour in the milk and beat for an additional 3 to 4 minutes.
To assemble: With a long serrated knife, split each cake layer horizontally into 2 layers. Place 1 layer on a cake plate and top generously with filling. Place another layer on top; top with more filling. Continue in this manner until you place the final layer of cake. Frost the cake all over with the buttercream. Serve at room temperature.
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