Add the sourdough starter, warm milk, flour, eggs, salt, sugar, and vanilla extract to the bowl of a stand mixer fitted with the dough hook.

Mix on low to medium speed until a shaggy dough forms.

With the mixer running, add the softened butter one tablespoon at a time, scraping down the bowl as needed between additions.

Increase the mixer speed to medium and knead for 7–10 minutes, or until the dough pulls away from the sides of the bowl and clings to the dough hook.

Note: Brioche dough is naturally very soft and sticky. Avoid adding extra flour, even if the dough feels wet.

Cover the bowl and let the dough bulk ferment at room temperature for 6–10 hours, or until the dough has risen about 30–50%. Timing will vary depending on the temperature of your kitchen.

Once bulk fermentation is complete, cover the bowl tightly with plastic wrap and refrigerate overnight.

Grease a 10-inch bundt pan thoroughly and set aside.

In a medium bowl, combine the granulated sugar and cinnamon.

Place the melted butter in a separate bowl.

Remove the chilled dough from the refrigerator and divide it into sections, then cut each section into small dough pieces.

Dip each dough piece into the melted butter, then coat generously in the cinnamon sugar mixture.

Arrange the coated dough pieces evenly in the prepared bundt pan.

Cover the pan and let the dough rise for 1–4 hours, or until visibly puffy and slightly expanded.

Preheat the oven to 350°F.

Add the brown sugar and butter to a small saucepan over medium heat.

Cook until the butter is melted and the sugar is mostly dissolved.

Remove from the heat and stir in the vanilla extract.

Pour the syrup evenly over the dough in the bundt pan.

Bake the monkey bread for 30–35 minutes, or until deeply golden brown and cooked through.

Allow the monkey bread to cool in the pan for 10 minutes before carefully inverting onto a serving plate.

In a small bowl, whisk together the powdered sugar, vanilla extract, and milk until smooth.

Drizzle the icing over the slightly cooled monkey bread before serving.