In a large mixing bowl, whisk together the flour, sugar, and salt, setting the mixture aside for later use.

Gently heat the milk and butter until the butter is melted and the mixture reaches around 110°F (43°C). Stir in the yeast until fully dissolved.

Combine the warmed milk mixture with the dry ingredients and add the egg. Stir using a strong spoon or mixer with a paddle attachment on medium speed until you form a soft dough.

Transfer the dough to a floured work surface and knead for 3 to 5 minutes until the dough is smooth, adding extra flour if it gets sticky. Place the dough in a greased bowl, cover lightly, and let it rest for 10 minutes.

Roll the dough into a rectangle measuring 14x8 inches, then spread the softened butter on top. Mix brown sugar and cinnamon, and sprinkle it evenly over the dough.

From the long side, tightly roll the dough into a log of approximately 14 inches. Cut the log into 10 to 12 equal sections and place them in a greased round or square baking dish.

Cover the dish with a towel or foil and allow the rolls to rise in a warm area for 60 to 90 minutes, until they have doubled in size.

Preheat your oven to 375°F (190°C). Bake the rolls for 24 to 27 minutes or until golden brown, covering with foil if they brown too quickly. The internal temperature should be between 195–200°F (91–93°C).

As the rolls slightly cool, beat the cream cheese in a bowl until it's creamy. Add softened butter and blend until smooth, then incorporate powdered sugar and vanilla until the mixture is well combined.

Spread the cream cheese icing over the warm rolls to enjoy immediately. Store any leftovers in an airtight container at room temperature for up to 2 days or refrigerate for up to 5 days.