Preheat oven to 325℉. Grease two 9-inch round cake pans, then line each with parchment paper (cut to fit), then grease the parchment. Flour the pans and set aside.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a small saucepan, heat together the milk and butter over low heat until the butter is melted. Remove the pan from the heat source and add the vanilla. Cover, set aside and keep warm.
Fit a stand mixer with the whisk attachment and whip the eggs and sugar together on high or until light and slightly thick, about 5 minutes.
Remove the mixing bowl from the stand and add the warm mixture, whisking by hand until blended. Add the flour and whisk again until incorporated.
Quickly divide the batter equally between the two pans and bake until the tops are light brown and a toothpick inserted in the center comes out clean, about 21 to 23 minutes. Rotate pans about halfway through baking.
Remove the cake from the oven and place the pans on a wire rack to cool complete. Once cool, run a small sharp knife around the edge of the pan and invert the cake onto the wire rack. Carefully peel away the parchment paper circle and turn the cakes right side up.
(Cooled cakes can be stored at room temperature for up to 1 day if wrapped tightly in plastic wrap. To freeze, wrap in plastic wrap, then foil and freeze for up to 1 month. Defrost at room temperature.)
While the cakes are cooling, prepare the pastry cream.
In a medium bowl whisk together the egg yolks, sugar and salt until smooth. Add the flour to the egg yolks and whisk again until incorporated. Set aside.
Heat the half-and-half in a medium saucepan over medium heat just until it simmers. Remove the half-and-half from the heat (leave the heat on) and, while whisking constantly, slowly add about ½ of the hot half-and-half to the yolks to temper the eggs. Whisking constantly, pour the tempered yolks back into the pan with the remaining half-and-half and whisk until combined.
Return the saucepan to the heat and cook, whisking constantly, until the mixture thickens, about 1 minute. Reduce the heat to medium-low and simmer, whisking constantly, for about 8 minutes.
Increase the heat to medium and cook the pastry cream, whisking vigorously, until bubbles burst on the surface, about 2-3 minutes.
Remove the pan from the heat and whisk in the butter and vanilla until the butter is melted and fully incorporated. If lumpy, strain the pastry cream through a fine-mesh strainer set over a clean medium mixing bowl. Lightly spray a piece of plastic wrap with vegetable cooking spray and place sprayed side down directly on top of the pastry cream in the bowl. Press lightly to insure all areas of the pastry cream are covered. Refrigerate for at least 2 hours or up to 24 hours.