In a large bowl, mix 2 cups of flour, sugar, salt, and yeast. In a microwave-safe bowl, heat the water, milk, and butter until warm (not hot). Add to the flour mixture, along with the egg, and mix until a soft dough forms. Add remaining flour as needed to handle.
Knead the dough on a floured surface for 4 minutes. Place in a greased bowl, cover, and let rise for 30 minutes.
Roll out the dough into a rectangle. Spread the softened butter over the dough, sprinkle with cinnamon, and top with crushed Oreo crumbs.
Roll up the dough and cut into 12 equal rolls. Place rolls in a greased 9×13 inch baking dish. Cover and let rise until doubled, about 30 minutes.
Preheat the oven to 375°F (190°C). Bake the rolls for 20-25 minutes until golden brown.
While the rolls bake, beat together cream cheese, butter, powdered sugar, and vanilla extract until smooth. Stir in Oreo crumbs. Spread over warm rolls before serving.
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