To the bowl of a stand mixer, add yeast and warm water and agitate. Let sit ten minutes until mixture is bubbly. In a saucepan, melt 1/3 cup (80 milliliters) butter with the milk. Remove from heat and let cool. Pour into bowl with yeast mixture and mix. Add eggs and whisk in. Add 1/3 cup (80 milliliters) sugar and mix in. Slowly add flour with the dough hook running. Stop adding flour when dough is tacky. Cover and let rise one hour, or until doubled. Melt the remaining butter in a saucepan. In a small bowl, mix together remaining sugar, brown sugar, and cinnamon. Pull off small pieces of dough and roll into a ball. Repeat until dough has been used up. When all dough balls have been formed, dip each one in the melted butter, then in the sugar mixture, and place in a buttered Bundt pan. Cover and let rise about 40 minutes. Bake at 350 F (180 C) for 30 minutes. Remove from oven. Let cool until just warm. Flip over onto a platter.
To make the glaze, mix together melted butter, icing sugar, milk, and vanilla. Pour glaze over top of monkey bread. Let cool slightly.