Step 1

For the banana sheet cake: Preheat the oven to 350 degrees F. Grease a 9-by-13-inch baking pan.

Step 2

Whisk together the cake flour, baking powder, salt and baking soda in a medium bowl. Beat the butter and granulated sugar at medium speed with a heavy-duty stand mixer until light and fluffy. Add the eggs, one at a time, beating just until the yellow disappears after each addition. Add the vanilla.

Step 3

Add the flour mixture alternately with the milk in three additions, beginning and ending with the flour. Add the bananas and beat just until combined.

Step 4

Pour the batter into the prepared pan and smooth the top. Bake until a toothpick inserted in the center comes out clean, 35 to 38 minutes. Let the cake cool in the pan for 10 minutes. Use the handle of wooden spoon (coated with cooking spray) or a straw to poke holes halfway through the cake in 1-inch intervals. Set aside.

Step 5

For the the pudding: Whisk together 1/4 cup milk with the cornstarch in a small bowl. Set aside. Combine the granulated sugar and salt with the remaining 1 3/4 cups milk in a medium saucepan over medium heat. Bring to a simmer and stir just until the sugar dissolves.

Step 6

Whisk the egg yolks in a medium bowl, then slowly add a little of the warm milk to the egg yolks, whisking constantly, until all the warm milk is incorporated. Whisk in the cornstarch mixture and return to the saucepan. Continue to cook over medium heat, whisking constantly, until the pudding thickens, about 10 minutes. Remove the pan from the heat and whisk in the butter and vanilla.

Step 7

To assemble: Pour the pudding over the cake, spreading evenly. Layer the banana slices over the pudding. Place plastic wrap directly over the surface of the bananas and pudding. Cover and refrigerate until the pudding is chilled, about 1 hour.

Step 8

Meanwhile, make the whipped cream. Combine the heavy cream, powdered sugar and vanilla in a large bowl. Beat with a handheld mixer until stiff peaks form. Spread the whipped cream over the cooled pudding and bananas. Garnish with more banana slices and crushed vanilla wafers, if desired.