To make the dough, put the warm water in a large bowl (or the bowl of your stand mixer) and sprinkle with the yeast and a pinch of the sugar. Let stand 5 minutes, or until foamy. (If it doesn’t foam, toss it and buy fresh yeast.)
In a small bowl, mix the warm milk and eggs together with a fork. Add to the yeast mixture along with 3 cups of the flour and the remaining sugar; mix until well blended and sticky. Add the butter and remaining flour and stir or beat with the dough hook attachment of your stand mixer until you have a soft, sticky dough; knead for about 8 minutes, until smooth and elastic. It will still be slightly tacky. Place back in the bowl, cover with a tea towel and let rise in a warm place for an hour, or until doubled in bulk.
Melt the butter in one bowl, and in another stir together the sugars and cinnamon. Pull the dough apart into balls the size of a small plum. Dip each ball of dough into the melted butter, then douse in the sugar mixture, rolling to coat completely. Place in a greased Bundt pan, piling them on top of each other.
Cover and let rise for an hour while you preheat the oven to 350F. Bake for 45 minutes, until golden and springy to the touch.
Invert onto a plate, then whisk together the icing sugar, milk and vanilla and drizzle overtop. Serve warm.