In a large bowl, stir together the warm milk, sugar and yeast. Let it sit for 5-10 minutes, until foamy.
Whisk in the melted butter, Greek yogurt and salt. Add the flour 1 cup at a time, stirring until a soft, shaggy dough forms.The yogurt stands in for the egg and keeps the bread extra soft and pillowy.
Knead the dough on a lightly floured surface for 6-8 minutes, until smooth, elastic and just slightly tacky. Place it in a greased bowl, cover and let it rise in a warm spot for about 1 hour, until doubled.
Grease a 10-cup bundt pan well. Mix the sugar and cinnamon in a bowl and pour the melted butter into a separate small bowl.
Punch down the dough, divide it into about 48 pieces, roughly 1 inch each, and roll them into balls. Dip each ball in the melted butter, roll it in the cinnamon sugar and layer the balls in the bundt pan.
In a small saucepan over medium heat, melt the butter with the brown sugar and any leftover cinnamon sugar. Let it bubble for 1 minute, stir in the vanilla and pour the caramel evenly over the dough.
Cover the pan and let the dough rise for 30-40 minutes, until puffy. Meanwhile, preheat the oven to 350°F.
Set the bundt pan on a baking sheet to catch any drips and bake for 30-35 minutes, until deep golden brown. Let it cool for 5 minutes, then flip it onto a serving plate and serve warm.