Line bottom of a 2-quart baking dish with half of the cookies. Top with half of the banana slices. Repeat layers.

Stir together brown sugar, cornstarch, and salt in a medium saucepan until no lumps remain. Stir in egg yolks and 1 teaspoon vanilla. Set pot over medium heat and gradually stir in milk, stirring constantly, until mixture becomes thick enough to coat the back of a metal spoon, 8 to 10 minutes. Remove from heat. Let cool 1 minute. Pour milk mixture evenly over bananas and cookies. Chill, covered, at least 2 hours or up to 12 hours.

Mix egg whites, cream of tartar, and remaining 1 teaspoon vanilla in a medium bowl with an electric mixer until soft peaks form. With mixer running, gradually add white sugar until stiff peaks form. Spread meringue over pudding.

Brown top of meringue with a kitchen torch, 2 to 3 minutes. (To store, insert toothpicks into meringue halfway up toothpick; loosely drape plastic wrap over toothpicks to cover. Chill up to 24 hours.)