In a small bowl, combine the warm milk, sugar, and yeast. Stir gently until the yeast dissolves and let it sit for about 5 minutes, or until it becomes bubbly and frothy.

In a large mixing bowl, combine the all-purpose flour and salt.

Add the activated yeast mixture, softened butter, egg, and vanilla extract into the flour mixture. Stir until just combined.

Lightly flour a clean surface and knead the dough for 5-7 minutes until it becomes smooth and elastic.

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

Preheat your oven to 375°F (190°C) and grease a 9x13-inch baking dish.

Once the dough has risen, punch it down to release the air. Roll it out into a rectangular shape, approximately 16x12 inches, on a floured surface.

Spread the melted butter evenly over the rolled-out dough. Sprinkle the brown sugar and cinnamon mixture over the buttered surface.

Starting from one long edge, roll the dough tightly into a log. Slice the log into 12 even pieces.

Place the cinnamon rolls into the prepared baking dish, ensuring they are snug against one another.

Bake in the preheated oven for 20-25 minutes or until the tops are golden brown and the rolls are cooked through.

While the rolls are baking, prepare the cream cheese icing by beating together the softened cream cheese, powdered sugar, vanilla extract, and milk until smooth.

Once the cinnamon rolls are baked and slightly cooled, generously spread the cream cheese icing over the top. Serve warm.