Set out a 9×13-inch baking dish (glass or ceramic). If you prefer a showy presentation, you can also use a large trifle bowl instead.
In a large mixing bowl, beat the softened cream cheese with a hand mixer until very smooth and fluffy, about 2 minutes. Scrape down the sides of the bowl as needed to remove any lumps.
8 oz cream cheese
Add the sweetened condensed milk, vanilla extract (if using), and a pinch of kosher salt to the cream cheese. Beat again until the mixture is silky, glossy, and completely combined.
14 oz sweetened condensed milk, 1 teaspoon pure vanilla extract, 1 pinch kosher salt
In a separate bowl, whisk the cold milk and instant vanilla pudding mix together for about 2 minutes, until it thickens slightly. It will continue to set as it stands.
2 cups cold whole milk, 1 box (5.1 oz) instant vanilla or French vanilla pudding mix
Gently pour the pudding mixture into the cream cheese mixture. Beat on low speed or whisk by hand until fully blended and smooth, with no visible lumps.
Add about two-thirds of the thawed whipped topping to the bowl and fold it in gently with a spatula until just combined and light. Reserve the remaining third of the whipped topping for the final layer.
12 oz frozen whipped topping
Arrange a single, even layer of Chessmen cookies or vanilla wafers across the bottom of the 9×13-inch dish, covering as much surface area as possible. Slight overlapping is fine.
2 packages Pepperidge Farm Chessmen cookies
Slice the bananas just before using. If making the dessert ahead, gently toss the slices with 1–2 teaspoons of lemon juice to help prevent browning. Spread the banana slices evenly over the cookie layer.
5-6 medium ripe bananas, 1-2 teaspoons lemon juice
Pour the creamy pudding mixture over the banana layer. Use a spatula to spread it into an even layer, making sure to nudge it into the corners and fully cover the bananas.
Spread the reserved whipped topping over the pudding in a smooth, even layer to create a fluffy top.
12 oz frozen whipped topping
Arrange another layer of Chessmen cookies or vanilla wafers on top of the whipped topping. If using wafers, you can stand some around the edges for a classic banana pudding look.
2 packages Pepperidge Farm Chessmen cookies
Cover the dish tightly with plastic wrap and refrigerate for at least 4 hours, or up to overnight. The longer it chills, the softer the cookies become and the more the flavors meld.
Slice into squares or scoop with a large spoon and serve cold. Garnish with extra cookie crumbs and fresh banana slices right before serving if desired.