In a large bowl, whisk together the flour, sugar, and salt. Set aside.
In a heatproof bowl or measuring cup, combine the milk and butter. Warm in the microwave or on the stove until the butter has melted and the mixture is warm to the touch (about 110°F/43°C, no higher). Whisk in the yeast until dissolved. Pour the mixture into the dry ingredients, add the egg, and stir with a sturdy rubber spatula or wooden spoon—or use a stand mixer fitted with the paddle attachment on medium speed—until a soft dough forms.
Transfer the dough to a lightly floured surface. Using floured hands, knead for 3–5 minutes until a smooth ball forms. If the dough is too soft or sticky, add a bit more flour as needed. Place the dough in a lightly greased bowl (non-stick spray works well), cover loosely, and let it rest for about 10 minutes while you prepare the filling.
After resting, roll the dough into a 14×8-inch rectangle. Spread the softened butter evenly over the surface. In a small bowl, mix the cinnamon and brown sugar, then sprinkle it evenly over the buttered dough. Starting from the long edge, roll the dough into a tight 14-inch log. Cut into 10–12 even rolls and arrange them in a lightly greased 9- or 10-inch round cake pan, pie dish, or square baking pan.
Cover the pan with aluminum foil, plastic wrap, or a clean kitchen towel. Let the rolls rise in a warm, draft-free spot for 60–90 minutes, or until doubled in size.
Once the rolls have doubled in size, preheat the oven to 375°F. Bake for 24–27 minutes, or until the tops are lightly browned. If they brown too quickly, tent the pan loosely with foil and continue baking. For precise doneness, the internal temperature should read 195–200°F on an instant-read thermometer. Remove from the oven and place the pan on a wire rack while you make the icing.
In a medium bowl, using a handheld or stand mixer fitted with a paddle or whisk attachment beat the cream cheese on high speed until smooth and creamy. Add the butter and mix until fully combined. Add the confectioners’ sugar and vanilla, and beat until smooth.
Spread the icing over the warm rolls using a knife or icing spatula. Serve immediately.
Cover leftover rolls tightly. Store frosted or unfrosted rolls at room temperature for up to 2 days or in the refrigerator for up to 5 days.
