Make the pudding: In a large saucepan, whisk together the sugar, cornstarch, and salt until combined.
Gradually whisk the milk into the dry mixture until smooth, then place the pan over medium-high heat and bring to a boil, scraping the bottom often.
When boiling, reduce heat to medium-low and continue to cook, stirring and scraping the bottom, until thick and large bubbles appear, about 4 to 6 minutes.
Remove from heat and whisk in the butter and vanilla until smooth. Transfer the pudding to a large bowl, press plastic wrap directly onto the surface, and refrigerate until cool, about 2 hours.
Make the stabilized whipped cream: Combine the water and gelatin in a small bowl and let soften about 5 minutes, then microwave until bubbling at the edges and gelatin dissolves, about 15–30 seconds.
Using an electric mixer, whip the 2 cups heavy cream on medium-low until foamy (about 1 minute), then on high until soft peaks form (about 2 minutes).
Pour the dissolved gelatin into the whipped cream and continue to whip until stiff peaks form, about 1 minute.
Fold the filling: Whisk one-third of the stabilized whipped cream into the chilled pudding to lighten it, then gently fold in the remaining whipped cream in about 1/4-cup additions until fully combined.
Assemble the cake: Line the bottom of a 9x13-inch baking pan with a layer of graham crackers, breaking as needed to fit.
Spoon half of the pudding-whipped cream mixture (about 5 1/2 cups) over the graham crackers and spread evenly, then add another layer of graham crackers.
Top with the remaining pudding mixture and finish with a final layer of graham crackers.
Make the chocolate glaze: In a microwave-safe bowl, combine the chocolate chips, 3/4 cup heavy cream, and corn syrup; microwave in 30-second intervals, stirring between, until smooth, about 1–2 minutes total.
Let the glaze cool to room temperature (about 10 minutes), then spread it over the top layer of graham crackers.
Refrigerate the assembled cake for at least 6 hours (or overnight) to set before slicing and serving.