To make the dough, combine the milk and the water, and warm the mixture to 110F. Then, stir in the sugar and yeast until the sugar has dissolved. Leave to stand for 5 minutes. Whisk in the melted butter and eggs.
In the bottom of your stand mixer combine 2 cups of the flour with the salt. Pour in the liquid mixture and use a rubber spatula to bring it together into a rough, sticky dough. Add the rest of the flour and bring everything together until just combined.
Fit the dough hook and knead the dough for 5 minutes; it should be smooth, elastic and slightly sticky. Cover and leave in a warm place to rise for 1 hour, or until the mixture has doubled in size.
To prepare the coatings, combine the brown sugar, sugar and cinnamon in one small bowl, and the melted butter and the vanilla in another.
Prepare a bundt pan by coating really well with softened butter or cooking spray, and preheat the oven to 350F.
Divide the dough into roughly 1 tbsp balls and using a fork, dip them first in the butter, then roll them in the sugar mixture, layering them in the prepared bundt pan as you go.
Bake the monkey bread for 45 minutes, covering the top with foil if you think the crust is getting too caramelized.
Remove from the oven and leave to cool in the pan for 15 minutes before turning out onto a wire rack.
To make the glaze, beat together the cream cheese, butter and vanilla until smooth and gradually add the powdered sugar until the mixture is thick and stiff.Gradually loosen it with the milk until you have reached the desired consistency. Drizzle it over the monkey bread once it has cooled.