Step 1: Prepare and Proof Dough: In a large bowl, whisk together your warm milk, active dry yeast, the egg, 1 tablespoon of melted butter, granulated sugar, and salt. Add 3 cups of all-purpose flour and stir gently until just combined (dough will be shaggy). Gradually add more flour, 1 tablespoon at a time, mixing after each addition, until the dough is no longer sticky. Turn onto a lightly floured surface and knead by hand for about 3 minutes until smooth and elastic. Return to the bowl, cover with plastic wrap, and let rise in a warm place for about 1 hour, or until doubled in size.
Step 2: Prepare Filling: While the dough rises, combine softened butter, packed brown sugar, and ground cinnamon in a small bowl. Mix well until it forms a smooth, spreadable paste. Set aside.
Step 3: Roll and Fill Dough: Once the dough has doubled, lightly flour a clean surface. Roll out the risen dough into an approximate 17x10 inch (43x25 cm) rectangle. Evenly spread the cinnamon filling across the entire surface, going all the way to the edges. Starting from one of the long edges, tightly roll the dough into a log. Slice the log into 8 equal cinnamon rolls using unflavored dental floss for clean cuts, or a sharp knife.
Step 4: Second Rise and Bake: Lightly grease a 9x13-inch (23x33 cm) baking pan. Arrange the 8 rolls in the pan, leaving a little space between them. Cover the pan with plastic wrap and let the rolls rise in a warm spot for another 30 minutes, or until visibly puffed up. While they rise, preheat your oven to 350°F (175°C). Once puffed, gently brush the tops of the rolls with the remaining 1 tablespoon of melted butter. Bake for 20-25 minutes, or until golden brown and fragrant. Remove from the oven and let cool in the pan for about 15 minutes.
Step 5: Prepare Frosting: While the rolls are cooling, combine softened butter, softened cream cheese, and vanilla extract in a medium bowl. Beat with an electric mixer until well combined and super smooth. Gradually add the powdered sugar, beating until the frosting is light, fluffy, and perfectly creamy.
Step 6: Frost and Serve: Spread the luscious cream cheese frosting evenly over the warm cinnamon rolls. The warmth will slightly melt the frosting, creating a gooey topping. Serve immediately.