Combine warm milk, yeast, and 1 teaspoon of the measured sugar. Let sit for 5–10 minutes until foamy.
Add remaining dough sugar, melted butter, egg, and salt.
Gradually add flour and mix until a soft dough forms.
Knead for 8–10 minutes by hand or 6–8 minutes using a stand mixer.
Place dough in a greased bowl, cover, and rise for 60–90 minutes until doubled.
Generously grease a 10–12-cup Bundt pan.
Mix cinnamon and granulated sugar for the coating in a bowl.
Place melted coating butter in a separate bowl.
Divide risen dough into approximately 40–48 small pieces.
Roll each piece into a ball, dip in melted butter, and coat with cinnamon sugar.
Arrange coated dough balls loosely in the prepared Bundt pan.
Heat brown sugar and butter in a saucepan over medium-low heat until combined. Remove from heat and add vanilla.
Pour the brown sugar sauce evenly over the dough pieces.
Cover and let rise for 30–45 minutes until puffy.
Preheat oven to 350°F (175°C).
Bake on a rimmed baking sheet for 35–45 minutes until golden and cooked through.
Cool in the pan for 8–10 minutes.
Carefully invert onto a heat-safe serving plate.
Beat cream cheese, powdered sugar, vanilla, and milk until smooth and pourable.
Drizzle glaze over slightly cooled monkey bread and serve warm.