Whisk the warm milk, yeast, and sugar in a bowl. Cover the bowl and set aside for 5 minutes.
Melt the butter and add to the yeast along with an egg. Whisk until smooth.
Fold in flour and salt. Stir until the dough comes together. Knead the dough on a lightly floured surface until smooth. Alternatively, just add it all to the stand mixer with a dough hook attachment.
Cover the dough with plastic foil and let it rise for one hour or until doubled in size.
Punch down the dough and cut it into 40-42 equal pieces. Shape each piece into a small ball.
Make the coating. In a bowl, combine the sugar with cinnamon. In a deep bowl, add half the melted butter. Generously grease the bundt pan with some butter.
Dip the dough balls into the butter, then coat them with the sugar-cinnamon mixture. Arrange the coated dough into the bundt pan.
Combine the remaining butter with brown sugar and vanilla extract. Drizzle the brown sugar mixture over the dough and let it rise for another 30 minutes.
Preheat oven to 180C/350F.
Bake the bread in the preheated oven for 35-40 minutes.
Cool the bread for 10 minutes before inverting it onto the plate. While the bread is cooling, prepare the icing by whisking together the confectioners' sugar with the milk and vanilla extract.
Drizzle the icing over the bread and serve immediately.
Notes
TO STORE: Leftover monkey bread can be stored at room temperature for up to 2 days. If you want it to last longer, refrigerate it in an airtight container for up to 1 week.TO FREEZE: Place slices of the monkey bread in a Ziplock bag and store it in the freezer for up to 6 months. Thaw overnight in the refrigerator.