To Make The Dough: In the bowl of a stand mixer fitted with the paddle attachment, combine warm milk, sugar, and yeast. Let stand until mixture is foamy, about 10 minutes.
With mixer running on low speed, add 1 cup (125 grams) flour, beating just until combined. 2. Add melted butter and ½ cup (63 grams) flour, beating until combined. Beat in eggs. Gradually add salt and remaining 1¾ cups (218 grams) flour, beating until a soft dough forms. (Dough will be sticky.)
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Loosely cover and let rise in a warm, draft-free place (75°F) until doubled in size, about 1 hour.
To make the coating: In one small bowl, combine the brown sugar and cinnamon. Stir to combine. In a second bowl, melt the butter.
Spray a 10-inch bundt pan with cooking spray.
Punch dough down very gently to release the gasses to release. Then remove small amounts of dough to form into about 1-inch balls. Dip each ball of dough into melted butter first, then roll into the brown sugar- cinnamon mixture. Place each ball of dough into the prepared pan arranging them as you go. You will get about 30 balls of dough. You may still see some of the bottom of the bundt pan in the end, but these will double in size. Once you have used all the dough, then cover the pan with a towel and allow to rise for another 45 minutes in a warm draft free place.
Pre-heat the oven to 350°F. Combine the rest of the brown sugar/cinnamon mixture and butter. You may need to re-melt the butter. Then pour the mixture over the monkey bread before baking. Bake in your pre-heated oven on the middle rack for 30-35 minutes. Insert a digital thermometer into the center. The internal temperature should read 190oF. Allow to cool in the pan for 5 minutes, before inverting onto a plate.
To make the glaze: In a bowl, whisk together the powdered sugar, milk and vanilla. If the glaze is too thick, adjust the consistency by adding additonal milk. Drizzle over the monkey bread and serve while warm.