In a large mixing bowl, whisk together the flour, granulated sugar, and salt. Set this dry mixture aside.

In a separate heatproof bowl, combine the milk and butter. Warm the mixture in the microwave or on the stovetop until the butter is fully melted and the liquid is warm to the touch (about 110°F/43°C—do not exceed this temperature). Stir in the yeast until fully dissolved.

Pour the warm yeast mixture into the dry ingredients, then add the egg. Using a sturdy wooden spoon or rubber spatula, stir until a soft dough begins to form. Alternatively, you can use a stand mixer fitted with a paddle attachment on medium speed to mix.

Turn the dough out onto a lightly floured surface and knead with floured hands for 3–5 minutes, until it becomes smooth and elastic. If the dough feels too sticky, add a bit more flour as needed. Shape into a smooth ball and transfer to a lightly greased bowl (non-stick spray works well). Cover loosely and let the dough rest for 10 minutes while you prepare the filling.

After resting, roll the dough into a 14×8-inch (36×20 cm) rectangle on a floured surface. Spread the softened butter evenly across the surface of the dough. In a small bowl, mix the brown sugar and cinnamon, then sprinkle this mixture evenly over the buttered dough.

Starting from the long edge, roll the dough tightly into a 14-inch log. Slice the log into 10–12 equal pieces. Arrange the rolls in a lightly greased 9- or 10-inch round cake pan, pie dish, or square baking dish.

Cover the pan loosely with plastic wrap, aluminum foil, or a clean kitchen towel. Let the rolls rise in a warm, draft-free place for 60 to 90 minutes, or until doubled in size. (For tips on creating the perfect rising environment, refer to a yeast baking guide.)

Once risen, preheat your oven to 375°F (190°C). Bake the rolls for 24–27 minutes, or until the tops are golden brown. If they start to brown too quickly, loosely cover the pan with foil and continue baking. To ensure doneness, you can use an instant-read thermometer—center rolls should reach an internal temperature of 195–200°F (91–93°C).

Remove the pan from the oven and place it on a wire rack. Prepare the icing while the rolls cool slightly.

In a medium bowl, beat the softened cream cheese using a hand mixer or a stand mixer fitted with a whisk or paddle attachment until smooth and creamy. Add the butter and beat until fully combined and smooth. Add the powdered sugar and vanilla, then continue beating until the icing is smooth and spreadable.

Spread the icing generously over the warm cinnamon rolls using a knife or icing spatula. Serve immediately for best texture and flavor.

Store leftover rolls (frosted or unfrosted) tightly covered at room temperature for up to 2 days, or refrigerate for up to 5 days.