In the bowl of a stand mixer, whisk together warm milk, egg, sugar, melted butter, vanilla and yeast together. Whisk in 1 cup flour.
Attach the dough hook to the stand mixer and add remaining 2 1/2 cups flour and salt. Knead the dough until it forms a ball. If the dough is sticking to the sides of the bowl, add more flour, 1 Tablespoon at a time.
Cover with plastic wrap and let rise in a warm place for 1 to 2 hours or until doubled in size. Meanwhile, generously grease a bundt pan with vegetable oil and set aside.
Punch dough down and transfer it to a lightly floured surface. Divide dough into quarters. Pinch off pieces of dough and roll into balls. Each quarter should yield about a dozen small dough balls (48 balls total).
Coating
In a small bowl, whisk together the brown sugar and cinnamon.
Dip each dough ball in melted butter, then roll in the cinnamon sugar until coated. Transfer to the prepared bundt pan. Repeat with remaining dough balls, tucking them in close to one another.
Cover the pan with plastic wrap or a damp cloth. Let the dough rise for 45 to 60 minutes or until doubled in size.
Preheat oven to 350℉. Bake monkey bread until golden brown, about 35 minutes. Cool in the pan for 5 minutes (see notes), then invert onto a wire rack set over a parchment lined baking sheet.
Glaze
For the glaze, whisk together powdered sugar, milk and vanilla. Drizzle over the warm bread and serve.
Notes
Active dry yeast: Substitute instant yeast for an equal amount of active dry yeast. Add warm milk, sugar and active dry yeast to the bowl and let it sit for 5 to 10 minutes. If it's foamy, then the yeast is good to use. Add remaining ingredients in step1 and continue with recipe.Cooling: Don’t let the bread much longer than 5 minutes in the pan or it will be hard to remove.Store cooled monkey bread tightly covered or in an airtight container for up to 2 days. Alternatively, it will last up to 5 days in the refrigerator.