01 - Preheat oven to 350°F. Grease and flour a 9x13-inch baking pan, ensuring even coverage on all sides and the bottom.
02 - In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly distributed.
03 - In a large mixing bowl, beat the room-temperature butter and granulated sugar with an electric mixer on medium-high speed until the mixture is pale, light, and fluffy, about 3 to 4 minutes.
04 - Add the eggs one at a time, beating well after each addition and scraping down the sides of the bowl as needed to ensure a uniform batter.
05 - Mix in the mashed ripe bananas and vanilla extract until fully incorporated into the batter.
06 - Alternately add the flour mixture and buttermilk to the wet ingredients in three additions, beginning and ending with the flour. Mix on low speed until just combined, being careful not to overmix.
07 - Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean. Allow the cake to cool completely in the pan.
08 - In a bowl, whisk together the instant vanilla pudding mix and cold milk for 2 minutes until the mixture thickens to a spreadable consistency.
09 - Spread half of the prepared pudding evenly over the completely cooled cake. Arrange the sliced bananas in a single layer on top, then spread the remaining pudding over the bananas, sealing them in.
10 - Refrigerate the cake for at least 30 minutes to allow the pudding layer to set firmly.
11 - While the cake chills, beat the heavy whipping cream, powdered sugar, and vanilla extract with an electric mixer on medium-high speed until soft peaks form. Spread or pipe the whipped cream evenly over the set pudding layer.
12 - Sprinkle the crushed vanilla wafer cookies over the whipped cream topping just before serving. Slice into portions and serve chilled.