Preheat oven to 350 F (177 C). Grease a 9 x 9-inch baking pan and line it with parchment paper, leaving an overhang on two sides for easy removal.
Pierce the sweet potato several times with a fork. Place it on a microwave-safe plate and microwave for about 8 minutes, turning halfway through cooking, until very tender. Allow it to cool slightly, then peel and mash until smooth. Measure 1 cup mashed sweet potato and set aside.
In a medium bowl, stir together the crushed graham crackers, ¼ cup unsalted melted butter, and 1 tablespoon light brown sugar until evenly moistened. Press the mixture firmly into an even layer in the bottom of the prepared pan.Bake for 5 minutes, then remove from the oven and allow to cool slightly. In a large bowl, whisk together the ½ cup hot melted butter and ½ cup unsweetened cocoa powder until smooth and glossy (Note 2).
Add the 1 cup mashed sweet potato, ½ cup granulated sugar, ½ cup light brown sugar, 2 large eggs, 1 tablespoon vanilla extract, and ½ teaspoon salt. Whisk until fully combined.
In a medium bowl, stir together the 1 cup all-purpose flour, ½ cup semi-sweet chocolate chips, and ½ cup mini marshmallows.
Fold the flour mixture into the wet ingredients until just combined. Do not overmix.
Spoon the batter over the cooled graham cracker crust and spread it evenly with a spatula.
Sprinkle the remaining 1 cup mini marshmallows over the batter. Arrange the ¾ bar milk chocolate evenly across the top, then finish with the ½ graham cracker sheet pieces.
Bake for about 40 minutes, until the edges are set, the center no longer appears glossy or wet, and the marshmallows are lightly toasted.Allow the brownies to cool completely in the pan before slicing into 9 squares (Note 3).