Generously coat the work surface with flour and turn the dough out onto the floured surface. Dust the top of the dough with more flour, then roll the dough into a 3/4-inch thick sheet. Use a 3-inch donut cutter to stamp out circles from the dough, then place the circles on a parchment paper-lined baking sheet. Bring the scraps together, re-roll them and stamp as many circles from the dough as possible (discard the remaining scraps); you should get about 12 doughnuts.
Line a rimmed baking sheet with paper towels. Add 2 to 3 doughnuts to the hot oil and cook until the doughnuts are golden brown, about 1 1/2 minutes on each side. Transfer to the paper towel-lined baking sheet to drain and cool and repeat with the remaining doughnuts, making sure the temperature of the oil returns to 350 degrees F between batches.
For the marshmallows: Return the oil to 350 degrees F. In a small bowl whisk together the flour, sugar, baking soda and salt. Add the milk, melted butter and vanilla and whisk until smooth. Dip a few marshmallows in the batter then add them to the hot oil (take care not to overcrowd the pot so they don't stick together) and fry until golden brown, about 30 seconds. Transfer the fried marshmallows to the paper towel-lined baking sheet and repeat with the remaining marshmallows.
For the glaze: Whisk together the confectioners' sugar and cocoa powder in a medium bowl, then add the milk and whisk until smooth. Dip one side of the cooled doughnuts into the glaze and return them to the baking sheet.
For the icing and for finishing the doughnuts: Place the marshmallows in a microwave-safe bowl and microwave on low power until they are melted (be careful-they melt quickly!). Stir until smooth, then immediately dollop a spoonful on top of each doughnut. Stick a fried marshmallow on top of the icing and serve.