Line the bottom and sides of a 12x12 or 13x9-inch baking pan with parchment or wax paper. Set aside.
Peel the potato and cut into equal, bite-sized chunks so that it cooks evenly. Bring water to a boil in a saucepan, add potatoes and cook for about 15 minutes.
Using a masher or food processor, mash hot potatoes with chocolate and butter. Mix in graham cracker crumbs, powdered sugar, cocoa powder, salt, vanilla, and rum until well combined.
Transfer the mixture into the prepared pan and press down to form a smooth surface.
In a small saucepan, heat the oil and butter over medium-high heat until butter is melted. Add chocolate and remove from the heat. Mix with a spoon or whisk until all chocolate is melted and smooth.
Pour glaze over the rum cake. Let sit at room temperature for 20 minutes, then refrigerate for at least an hour.
Flip the cake out of the pan onto a clean surface and remove the parchment paper. Slice into bars.
Prep the Ingredients - Use a good quality chocolate bar for cooking, like Ghirardelli. Chop the chocolate bar before you cook potatoes. You don't want to have to do that when potatoes are done because it needs to melt while potatoes are hot. Measure the graham cracker crumbs, sugar, cocoa, and the rest of the ingredients while potatoes are cooking.
Use your hands, a bottom of a glass, or an offset spatula to press down potato chocolate rum mixture into the prepared pan.
For a stronger rum flavor, use a teaspoon of rum extract.
These DIY chocolate bars are best cut when they are well chilled.
For the mess-free cutting of the cake into bars, use a hot knife. To do this, fill a tall container/pitcher with hot tap water deep enough to cover the entire blade of your knife. Dip the knife into the hot water, and wipe it dry on a clean towel before making a cut.
You can use any kind of starchy potato, but Russet potato works the best.
Store refrigerated in an air-tight container.
For a different shape, roll the mixture into balls and dip into chocolate glaze or roll into desiccated coconut.