1. Whisk together first 4 ingredients in a large saucepan over medium-low heat. Cook, whisking constantly, 15 to 20 minutes or until thickened. Remove from heat; stir in butter and vanilla until butter is melted.
2. Fill a large bowl with ice. Place saucepan in ice, and let stand, stirring occasionally, 30 minutes or until mixture is thoroughly chilled.
3. Meanwhile, cut bananas into 1/4-inch slices. Break cookies into thirds.
4. Spoon half of pudding mixture into a 3-qt. bowl or pitcher. Top with bananas and cookies. Spoon remaining pudding mixture over bananas and cookies. Top with sweetened whipped cream. Cover and chill 2 to 24 hours. Garnish, if desired.
Note: For testing purposes only, we used Nabisco Nutter Butter® Sandwich Cookies. For photography, we divided mixture between 2 (1 1/2- to 2-qt.) wide-mouthed pitchers.
Shortcut Nutter Butter®-Banana Pudding Trifle: Omit eggs, sugar, flour, and butter. Substitute thawed extra creamy whipped topping for sweetened whipped cream. Reduce vanilla to 1 tsp. Place 3 cups milk and vanilla in large bowl; add 2 (3.4-oz.) packages vanilla instant pudding mix. Beat with an electric mixer at medium speed 2 minutes or until thickened; let stand 5 minutes. Stir in 1 (8-oz.) container sour cream. Proceed with recipe as directed in Steps 2 through 4.
Note: For testing purposes only, we used Jell-O Vanilla Instant Pudding and Pie Filling and Cool Whip Extra Creamy.