Preheat the oven to 350°F and line a 9x13-inch baking pan with parchment paper, leaving an overhang to lift the bars out later.
Combine the graham cracker crumbs, melted butter, and sea salt in a medium bowl. Stir until the crumbs are evenly moistened and hold together when pressed.
Press the crumb mixture firmly into the bottom of the prepared pan using the bottom of a measuring cup or a flat glass to create an even layer.
Bake the crust for about 12 minutes, until fragrant and slightly set. Remove from the oven and let cool on a wire rack while you prepare the fillings.
In a large bowl, beat the room-temperature cream cheese with the powdered sugar and vanilla until completely smooth and lump-free, using an electric mixer on medium speed.
Fold about 2 cups of the whipped topping into the sweetened cream cheese gently with a rubber spatula until light and fluffy. Reserve the remaining whipped topping for the final layer.
Spread the cream cheese mixture evenly over the cooled graham crust, smoothing the top with an offset spatula or the back of a spoon.
Whisk the cold milk and the chocolate pudding mix in a bowl for about 2 minutes, or until the pudding begins to thicken and is free of lumps (follow manufacturer timing if different).
Pour the chocolate pudding gently over the cream cheese layer and spread it to the edges so it forms an even middle layer.
Top the pudding with the remaining whipped topping, smoothing the surface. Sprinkle mini chocolate chips or shaved chocolate on top if using.
Refrigerate the assembled pan for at least 4 hours (preferably overnight) until the pudding and cream layer are fully set. Use the parchment overhang to lift the slab from the pan and cut into 12 bars.