Combine the dry ingredients (the first 3 ingredients) in a large saucepan, whisking to break up any lumps.
Separately, whisk the eggs and then add them in. Add the milk and stir until smooth.
Turn the stovetop heat on medium-low and cook the pudding until it's nice and thick, for about 15 to 25 minutes. Stir it constantly to avoid it from scorching the bottom or turning lumpy. If it bubbles, turn the heat down.
Remove it from the heat and add the butter substitute and baking extract, mixing to combine. Allow the pudding to sit at room temperature for a few minutes, so it is just warm instead of hot.
In an 8x8 square pan, 9-inch round, or smaller dish, assemble the banana pudding with fresh-sliced banana pieces and vanilla wafer cookies. I usually put the slices from one banana on the bottom, top it with the pudding, and then create a design with the remaining banana pieces and the cookies, but feel free to get creative and layer it how you want!
Serve immediately to enjoy it warm or chill in the refrigerator until it’s more set and cold, depending on your preferences.