Melt the butter in a saucepan over medium heat or microwave in 15-second intervals. Add the milk and microwave in 30-second intervals or cook in the saucepan until it's just lukewarm, not hot. Pour the milk and melted butter into a large bowl or the bowl of a stand mixer, stir in the sugar and the yeast and let sit until the yeast is dissolved and slightly foamy.
Stir in the salt and flour and mix until a dough is formed. When measuring the flour, loosen it up in the bag to fluff it up then lightly spoon it into your measuring cup and level it off with the flat side of a knife. Start with 5 cups of flour, if the dough is too sticky add 1 tbsp. more.
Knead the dough for about 5 minutes in the stand mixer fitted with a paddle attachment or knead with your hands on a lightly floured surface. To knead with your hands take the dough and push it away from you with the palm of your hand, then pull it back towards you and repeat. The dough should be soft, not too sticky, and bounce back when you make a gentle indent with your finger.
Form the dough into a ball and place it in a large bowl that has been lightly greased with butter. Cover the bowl with a kitchen towel or plastic wrap and let rise for about 1 hour or until doubled in size.
Shape the dough: Punch the dough down to release the air. Tear off 1 and ½ tablespoons of dough and roll it into a ball with the palm of your hands. Place it on a large baking sheet lined with parchment paper and repeat with the rest of the dough.
Cinnamon-Sugar Coating: Melt ½ cup of unsalted butter in one bowl and in a separate bowl whisk granulated sugar, brown sugar, and cinnamon. Dip each dough ball into the melted butter and then toss with the cinnamon-sugar coating.
Preheat your oven to 350°F.
Generously spray a 9 or 10″ bundt pan with non-stick spray or grease it with butter. Arrange the dough balls around the outer edge of the pan. (See post for a visual)
Cinnamon-Sugar Sauce: Melt ¼ cup of unsalted butter and whisk in the ½ cup brown sugar and ½ tbsp. ground cinnamon. Spread the sauce all over the dough in the pan.
Bake: Bake the monkey bread for 20-22 minutes.
Once the pan is cool enough to hold, cover it with a plate and flip the pan over to invert the monkey bread onto it. Don’t let it cool for too long or the bread won’t come out of the pan as easily.
Make the Icing: Whisk the powdered sugar, milk, vanilla extract, and salt in a medium bowl or a glass measuring cup until smooth. If the icing is too thick you can add more milk, or if it’s too thin you can add more powdered sugar.
Pour the icing over the monkey bread and serve warm.