Prepare the yeast: Warm the milk until comfortably warm to the touch. Mix it with the active dry yeast and a little of the measured sugar. Let it stand for 5–10 minutes, until foamy.
Make the dough: Combine the flour, remaining sugar, and salt in a large mixing bowl. Add the yeast mixture, egg, and melted butter. Mix until a soft dough forms, then knead for 8–10 minutes by hand or 5–7 minutes with a dough hook, until smooth and elastic.
Let the dough rise: Place the dough in a lightly greased bowl and cover it. Let it rise in a warm place for 60–90 minutes, or until roughly doubled in size.
Prepare the cinnamon sugar: Mix the granulated sugar and ground cinnamon in a shallow bowl. Place the melted butter in a separate bowl for coating the dough.
Shape the dough balls: Gently punch down the risen dough and divide it into small, evenly sized pieces. Roll each piece into a ball, dip it in melted butter, and coat it in the cinnamon-sugar mixture.
Arrange in the Bundt pan: Grease a standard Bundt pan thoroughly. Arrange the coated dough balls loosely in the pan, layering them evenly. Cover and let rise for another 30–45 minutes, until puffy.
Make the brown sugar glaze: Melt the butter in a small saucepan over low heat. Stir in the brown sugar and vanilla until combined and the sugar begins to dissolve. Remove from the heat.
Add the glaze: Pour the brown sugar mixture evenly over the dough balls, allowing it to flow between the layers.
Bake the monkey bread: Preheat the oven to 350°F (175°C). Bake for 30–40 minutes, until golden brown and cooked through in the center. If the top browns too quickly, loosely cover it with foil.
Cool and serve: Let the monkey bread rest in the pan for 5–10 minutes. Carefully invert it onto a serving plate and lift away the pan. Serve warm while the bread is soft and the glaze is sticky.