Whisk the granulated sugar, cornstarch, egg yolks, vanilla, and salt in a saucepan. Gradually whisk in the oat milk and coconut milk until smooth.
Cook the pudding over medium heat, whisking constantly, until it bubbles and thickens. Continue cooking for 1 minute, then transfer it to a bowl.
Press parchment paper directly against the pudding surface. Refrigerate for at least 3 hours, then whisk briefly until creamy.
Cut the bananas into thick coins. Melt the vegan butter with the dark brown sugar, cinnamon, and salt in a skillet over medium heat.
Add the banana coins and cook for about 1 minute per side, until coated in glossy caramel but still holding their shape. Cool completely.
Layer vanilla wafers, chilled pudding, caramelized bananas, and whipped topping in 6 dessert glasses. Repeat the layers, reserving a few cookies and bananas for the tops.
Refrigerate for 2 hours to soften the cookies slightly. Finish with the reserved cookies and caramelized bananas just before serving.
