Spray a Le Creuset dutch oven (about 4.5 quarts) very well with nonstick spray.
In a large bowl, mix together the warm milk, warm water, melted butter, sugar, and yeast. Let sit for 5 minutes until foamy.
In a stand mixer fitted with the dough hook, combine the flour and salt. With the mixer on low, slowly pour in the wet ingredients. Once the dough comes together, increase speed to medium and knead until shiny and smooth, 6 to 7 minutes.
Transfer the dough to a greased bowl, cover, and let rise in a warm place until doubled in size, 50 to 60 minutes.
In one bowl, stir together the brown sugar and cinnamon. Place the melted butter in a second bowl.
Turn the risen dough onto a lightly floured surface and pat it into an 8 inch square. Cut into 64 pieces and roll each piece into a small ball. Dip each dough ball in melted butter, then roll in the cinnamon sugar mixture. Layer the coated balls in the dutch oven, staggering seams to ensure even baking.
Cover and let rest until puffy, about 45 minutes.
Preheat the oven to 350 degrees. Bake for 35 to 40 minutes until the top is deep golden brown and caramel is bubbling at the edges.
Cool in the pan for 5 to 10 minutes, then carefully invert onto a serving platter.
While the monkey bread cools slightly, beat the cream cheese until smooth. Add the powdered sugar, milk, and vanilla, whisking until pourable and lump free. Drizzle over the warm monkey bread and serve immediately.