Preheat the oven to 350° F. Grease an 8-inch round cake pan with non-stick spray. Line the bottom with parchment paper and spray the bottom of the parchment paper with non-stick spray.
In a small mixing bowl, whisk together the graham cracker crumbs, flour, baking powder and salt. Set aside.
Place the eggs and the sugar in the mixing bowl of a stand mixer. With the whisk attachment, beat the eggs and sugar on medium speed for 5 minutes, then increase to high speed for an additional 4-5 minutes. The mixture will be pale and thick and will hold its shape for at least several seconds when dropped on itself.
While the eggs and sugar are whisking, combine the milk and butter in small saucepan and heat over low heat, stirring occasionally to help the butter melt, just until the mixture comes to a simmer. Turn the heat off and stir in the vanilla.
Set the mixer to the lowest speed and add the dry ingredients, just until almost fully combined. With the mixer still running on the lowest speed, slowly pour the hot milk mixture into the batter and continue mixing just until all ingredients are fully combined. Use a spatula to scape the sides of the bowl and to ensure the batter is fully mixed. Pour the batter into the prepared pan.
Bake for 33-36 minutes. The cake should be lightly brown on top and a toothpick inserted in the center should come out clean. Allow the cake to cool in the pan for 15 minutes before using a metal spatula or knife to release it from the sides of the pan. Invert the cake onto a wire rack to cool completely.