Gather all ingredients.

Combine sugar, flour, and salt together in a medium saucepan. Whisk in eggs and milk; cook mixture over low heat, stirring constantly (and patiently), until mixture has thickened enough to coat the back of a spoon, up to 15 minutes.

Remove from heat and continue to stir until pudding has cooled slightly; whisk in butter and vanilla until incorporated and smooth.

Spread pudding into a large serving dish; layer with sliced bananas and vanilla wafers.

Chill at least 1 hour in the refrigerator before serving.