In a medium saucepan, whisk together the sugar, cornstarch, and salt. Gradually whisk in the oat milk until smooth, then whisk in the egg yolks.
Cook over medium heat, whisking constantly, for 6 to 8 minutes, until the mixture thickens and begins to bubble. Keep whisking for 1 more minute, then remove from the heat and whisk in the butter and vanilla.
Transfer the pudding to a bowl, press plastic wrap directly onto the surface, and refrigerate for about 2 hours, until completely cold.
Whisk the cold pudding briefly to loosen it, then gently fold in the whipped topping in two additions until smooth and fluffy.
Line the bottom of a 9x13-inch dish with a single layer of graham crackers, breaking them as needed to fit. Spread half of the filling over the crackers, add a second layer of crackers, spread the remaining filling, and finish with a final layer of crackers.
Heat the coconut milk until steaming and pour it over the chocolate chips and butter in a bowl. Let it sit for 5 minutes, stir until glossy and smooth, then cool for 10 minutes and spread it evenly over the top layer of crackers.
Cover and refrigerate for at least 8 hours or overnight, until the graham crackers soften into tender, cake-like layers. Slice and serve cold.