In a medium saucepan, combine 3 cups whole milk and 3 tablespoons granulated sugar. Warm over medium heat, stirring occasionally, until steam rises and small bubbles form at the edges (do not boil).

Temper eggs

In a bowl, whisk together 4 egg yolks, 2 tablespoons cornstarch, and a pinch of salt until smooth and slightly pale. Slowly pour about 1/2 cup of the hot milk into the yolk mixture while whisking constantly to temper the eggs.

Thicken custard

Pour the tempered egg mixture back into the saucepan with the remaining hot milk, whisking constantly. Cook over medium-low heat, stirring continuously, until the mixture thickens to a pudding-like consistency (coats the back of a spoon), about 3–5 minutes. Remove from heat.

Finish custard

Stir in 2 tablespoons unsalted butter and 2 teaspoons vanilla extract until fully incorporated. Taste and add a pinch more salt if needed. Transfer the custard to a clean bowl and press plastic wrap directly onto the surface to prevent a skin from forming. Chill until cool, about 1–2 hours (or 30 minutes in the freezer, stirring once).

Whipped Cream

Chill a medium mixing bowl and beaters for 10 minutes in the freezer. Pour 1 cup cold heavy cream into the chilled bowl, add 2 tablespoons powdered sugar and 1/2 teaspoon vanilla extract. Whip on medium-high speed until soft peaks form, about 2–3 minutes. Do not overwhip. Fold one-third of the whipped cream into the cooled custard to lighten it, then gently fold in the remaining whipped cream until smooth and uniform.

Assembly

Prepare bananas

Peel and slice 4 ripe bananas into 1/4-inch slices. To prevent excessive browning, toss slices lightly with a squeeze of lemon juice if desired (lemon not included in ingredient list).

Layer pudding

In a 9x13-inch dish (or individual serving glasses), spread a thin layer (about 1 cup) of the custard on the bottom. Arrange a single layer of vanilla wafers (about 12) over the custard. Top the wafers with half of the banana slices (about 2 bananas). Spread about one-third of the remaining custard over the bananas. Repeat layers: another layer of wafers (about 12), the remaining bananas, and another third of the custard. Finish with a final layer of wafers and the remaining custard, smoothing the top.

Chill to set

Cover the assembled pudding with plastic wrap and refrigerate for at least 4 hours, preferably overnight, to allow the wafers to soften and flavors to meld.

Garnish and serve

Just before serving, whip the remaining reserved whipped cream if any (or use any remaining dollops) and spread or pipe over the pudding. Sprinkle 1/4 cup crumbled vanilla wafers and up to 1/2 cup extra banana slices on top as garnish. Slice and serve chilled.