In a small bowl, combine warm milk, yeast, and 1 tablespoon of the granulated sugar. Let sit for 5-10 minutes until foamy.
In a large bowl, combine warm water, the remaining granulated sugar, flour, melted butter, egg, and salt. Add the yeast mixture.
Mix until a dough forms. Knead on a lightly floured surface for 5-7 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm place for 1-1.5 hours, or until doubled in size.
While the dough is rising, prepare the topping by combining brown sugar and cinnamon in a shallow dish. Melt the butter for the topping in another shallow dish.
Once the dough has risen, punch it down. Roll it into a log and cut into 1-inch pieces. Dip each piece into the melted butter, then into the brown sugar mixture.
Arrange the coated dough pieces in a greased bundt pan or 9x13 inch baking dish.
Cover the pan and let the dough rise again for 30 minutes.
Preheat oven to 350°F (175°C). Bake for 30-35 minutes, or until golden brown.
While the bread is baking, whisk together the powdered sugar, milk, and vanilla extract for the glaze.
Let the monkey bread cool for 10 minutes in the pan, then invert onto a serving plate. Drizzle with glaze.