egin by preheating the oven to 350°F. Prepare a 9-inch round cake pan by spraying it with baking spray that contains flour. Additionally, line the bottom of the pan with parchment paper and give it another spray.
In a medium saucepan, combine the brown sugar, brewed coffee, chocolate chips, and vegetable oil. Heat the mixture over medium heat while stirring constantly until the sugar has completely dissolved and the chocolate is fully melted. Once melted, remove the saucepan from the heat and allow it to cool for approximately 10 minutes.
In a separate medium bowl, whisk together the all-purpose flour, dark cocoa powder, baking soda, and kosher salt until well combined.
Gradually incorporate the eggs, sour cream, and vanilla extract into the cooled brown sugar mixture, whisking each ingredient in one at a time. Slowly add the dry flour mixture, whisking just until all ingredients are combined without overmixing.
Pour the batter into the prepared cake pan. Bake in the preheated oven for 35 to 40 minutes, or until a wooden toothpick inserted into the center comes out with a few moist crumbs. After baking, let the cake cool in the pan for 10 minutes. Carefully turn the cake out onto a plate, remove the parchment paper, and then flip it back onto a wire rack, allowing it to cool completely, top-side up.
For the frosting, melt the salted butter in a medium heatproof bowl, approximately 30 seconds in the microwave. Once melted, whisk in the dark cocoa powder and vanilla extract. Using an electric mixer on low speed, gradually mix in the powdered sugar, approximately 1 cup at a time, alternating with the hot water, about 2 tablespoons at a time, until well combined. Increase the mixer speed to medium and beat the mixture until it is smooth and spreadable, taking about 1 to 2 minutes.
Once the cake has completely cooled, place it on a serving plate and generously spread the frosting over the top and sides. Chill the cake until the frosting has set, which can be accomplished by refrigerating it for 30 minutes or placing it in the freezer for 15 minutes.
In a small heatproof bowl, microwave the marshmallows until they puff up and become melted in the center, roughly 15 seconds.
Using two forks or your fingers (ensuring you wear food-safe gloves), dip them in the melted marshmallows and stretch them until they resemble thin webs. Wrap these marshmallow strands around the cake, repeating the process until the entire cake is covered. Allow the marshmallows to set for at least 10 minutes before slicing and serving.