Step 1: Activate the Yeast In a large bowl or the bowl of a stand mixer, combine the warm milk and one tablespoon of the sugar. Sprinkle the yeast over the top and let it sit for about 5 to 10 minutes until it becomes foamy and fragrant.
Step 2: Mix the Dough Add the remaining sugar, melted butter, salt, and egg to the yeast mixture. Whisk until combined. Gradually add the flour, one cup at a time, mixing with a dough hook or wooden spoon until a soft dough forms.
Step 3: Kneading the Dough Knead the dough for about 5 to 7 minutes on a lightly floured surface or in your stand mixer. The dough should be smooth, elastic, and slightly tacky to the touch but not sticking heavily to your fingers.
Step 4: The First Rise Place the dough in a lightly oiled bowl and cover it with a clean kitchen towel. Let it rise in a warm, draft-free spot for about 60 to 90 minutes, or until it has doubled in size.
Step 5: Roll and Fill Punch down the risen dough and turn it out onto a floured surface. Roll it into a large rectangle (about 12×18 inches). Spread the softened butter over the dough, then sprinkle evenly with the brown sugar and cinnamon mixture.
Step 6: Shape the Rolls Starting from the long edge, roll the dough up tightly into a log. Use a sharp knife or unflavored dental floss to cut the log into 12 even pieces. Place the rolls into a greased 9×13 inch baking pan.
Step 7: The Second Rise Cover the pan and let the rolls rise again for 30 to 45 minutes. During this time, preheat your oven to 350°F (175°C).
Step 8: Baking the Rolls Bake the rolls for 20 to 25 minutes, or until they are lightly golden brown on top. Avoid overbaking to keep them soft and gooey.
Step 9: Prepare the Frosting While the rolls bake, beat together the cream cheese and butter until smooth. Add the powdered sugar, vanilla, and milk, mixing until creamy and spreadable.
Step 10: Glaze and Serve Allow the rolls to cool for about 5 to 10 minutes, then spread the cream cheese frosting generously over the warm rolls. Serve immediately while they are fresh and soft.