In a large mixing bowl, whisk together the flour, sugar, salt, and yeast. In a small saucepan, heat the milk, water, and butter over low heat until the butter is melted and the mixture is warm to the touch (around 110°F).

Add the milk mixture and egg to the flour mixture and stir until a soft dough forms. Knead the dough on a floured surface for 5-7 minutes, or until it is smooth and elastic.

Place the dough in a greased bowl, cover it with a damp towel, and let it rise in a warm, draft-free place for about 1 hour, or until it has doubled in size.

In a medium mixing bowl, stir together the softened butter, brown sugar, and cinnamon until well combined.

Once the dough has risen, punch it down and roll it out on a floured surface into a large rectangle (about 16 x 12 inches).

Spread the filling evenly over the dough, leaving a small border around the edges. Roll the dough up tightly, starting from the long side, and pinch the edges to seal.

Cut the rolled dough into 12 even slices using a sharp knife or a piece of dental floss. Place the slices in a greased 9 x 13 inch baking dish, cover with a damp towel, and let them rise in a warm, draft-free place for about 30 minutes, or until they are puffy and have doubled in size.

Preheat the oven to 375°F. Bake the cinnamon rolls for 20-25 minutes, or until they are golden brown and puffy. Remove them from the oven and let them cool slightly.

In a small mixing bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.

Once the rolls have cooled slightly, drizzle the glaze over the top of them using a spoon or a pastry bag. Serve the cinnamon rolls warm, with a hot cup of coffee or tea.