In a bowl, whisk together the warm milk, yeast, and sugar. Cover the bowl and set it aside for 5 minutes.

Melt the butter and add it to the yeast mixture along with the egg. Whisk until smooth.

Fold in the flour and salt. Stir until the dough comes together. Knead on a lightly floured surface until smooth, or use a stand mixer with a dough hook attachment.

Cover the dough with plastic wrap and let it rise for an 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.

For the coating, mix the sugar with cinnamon in a bowl. Melt half of the butter in a deep bowl. Grease a bundt pan with some butter.

Dip each dough ball into the melted butter, then coat them with the sugar-cinnamon mixture. Arrange the coated dough balls in the bundt pan.

Mix the remaining melted butter with brown sugar and vanilla extract. Drizzle this mixture over the dough and let it rise for another 30 minutes.

Preheat the oven to 350F.

Bake the bread in the preheated oven for 35-40 minutes.

Cool the bread for 10 minutes before inverting it onto a plate. While it cools, prepare the icing by whisking together the confectioners sugar with milk and vanilla extract.

Drizzle the icing over the bread and serve immediately.