Place 3 cups milk and vanilla in large bowl; add vanilla instant pudding mix. Beat with an electric mixer at medium speed 2 minutes or until thickened; let stand 5 minutes. Stir in sour cream.
Fill a large bowl with ice. Place bowl in ice, and let stand, stirring occasionally, 30 minutes or until mixture is thoroughly chilled.
Meanwhile, cut bananas into 1/4-inch slices. Break cookies into thirds.
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 whipped topping. 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.
Note: For testing purposes only, we used Jell-O Vanilla Instant Pudding and Pie Filling and Cool Whip Extra Creamy.