Place milk into a measuring cup and microwave for one minute, until warm. Mix active dry yeast and 1 teaspoon granulated sugar into the milk. Let sit for 10 minutes.
In a large bowl, combine flour, remaining sugar and salt. Add in yeast mixture, melted butter and eggs. Mix until a dough forms. Knead dough out onto a floured surface, adding more flour as needed until smooth.
Place dough into a large greased bowl. Cover and let rise for 1-2 hours, or until doubled in size. Combine brown sugar and cinnamon in a shallow dish. Place melted butter into another shallow dish.
Preheat oven to 350 F. Deflate dough and divide into 40-45 small balls. Roll each ball into the melted butter, then coat in the cinnamon sugar mixture. Place coated balls into a greased bundt pan. Cover and let rise at room temperature for another 30-60 minutes. Or refrigerate for at least 8 hours or overnight. When ready to bake, let rise at room temperature for 30-60 minutes.
Bake uncovered for 35-40 minutes. Let sit in pan for 5 minutes. Turn pan upside down onto a plate. For the frosting, whisk powdered sugar, milk and vanilla extract together until smooth. Drizzle frosting over monkey bread.
