This recipe combines the familiar flavors of pumpkin pie and a classic layer cake. Buttery vanilla cake layers flavored with pumpkin spice sandwich a decadent pumpkin pudding, then the cake is enrobed in a luxurious sweet whipped cream frosting. A perfect marriage of cake and pie.
Make the pumpkin filling: Combine the milk, pumpkin, granulated sugar, cornstarch, egg yolks, vanilla, pie spice and salt in a medium saucepan. Cook over medium heat, whisking occasionally, until bubbles start to form around the edges, about 5 minutes. Continue cooking, whisking constantly and getting into the corners of the saucepan, until the mixture thickens to a pudding-like consistency, about 5 more minutes. Transfer to a heatproof bowl; press plastic wrap directly on the surface and refrigerate until cooled, 1 1/2 to 2 hours.
Meanwhile, make the cake: Preheat the oven to 325˚ F. Butter two 9-inch round cake pans; line the bottoms with parchment paper and butter the paper. Lightly dust the pans with flour, tapping out the excess. Combine the flour, baking powder, pie spice and salt in a large bowl and whisk to combine.
Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the eggs, one at a time, then beat in the vanilla. Add the flour mixture in three batches, alternating with the milk in two batches; beat until just combined.
Divide the batter between the 2 pans. Bake until light golden brown on top and the center springs back when gently pressed, 30 to 35 minutes. Transfer to a rack and let cool 10 minutes in the pans, then remove the cakes to the racks to cool completely. Remove and discard the parchment.
Put 1 cake layer on a cake stand or platter. Spread the pumpkin filling on top, leaving a 1/2-inch border. Top with the second cake layer.
Make the whipped cream: Combine the heavy cream, confectioners' sugar, milk powder and vanilla in a large metal bowl (for the best results, chill the bowl beforehand). Beat with a mixer on medium-high speed until medium peaks form, 3 to 4 minutes. Spread the whipped cream all over the cake and refrigerate until ready to serve, at least 20 minutes. Sprinkle with pumpkin pie spice before serving.