In a large pot, combine the mashed bananas, sugar, and water. Heat the mixture over medium heat, stirring occasionally, until the mixture reaches a simmer.
Reduce the heat to low and add the vanilla pudding mix. Stir continuously until the pudding mix has dissolved completely and the mixture has thickened slightly.
Slowly pour in the whole milk while stirring constantly. Continue to heat the mixture over low heat until it reaches a gentle simmer.
Remove the pot from the heat and let the mixture cool for about 15 minutes.
Once the mixture has cooled, stir in the grain alcohol or moonshine and vanilla extract. Mix well to ensure everything is combined.
Using a funnel, carefully pour the banana pudding moonshine into clean mason jars, leaving about 1 inch of headspace at the top.
Lid tightly the jars and put them in a cool and dark place for two weeks, allowing the flavors to meld together.
After the aging period, strain the moonshine through cheesecloth or coffee filters to remove any solids. Discard the solids and transfer the strained liquid back into the jars.
Seal the jars tightly and refrigerate until ready to serve.