1Make the dough: Whisk the flour, sugar, and salt together in a large bowl. Set aside.
2
Combine the milk and butter together in a heatproof bowl. Microwave or use the stove and heat 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 it has dissolved. Pour mixture into the dry ingredients, add the egg, and stir with a sturdy rubber spatula or wooden spoon OR use a stand mixer with a paddle attachment on medium speed. Mix until a soft dough forms.
3
Transfer dough to a lightly floured surface. Using floured hands, knead the dough for 3 minutes. You should have a smooth ball of dough. If the dough is super soft or sticky, you can add a little more flour. Place in a lightly greased bowl (I use non-stick spray), cover loosely, and let the dough rest for about 10 minutes as you get the filling ingredients ready.
4Fill the rolls: After 10 minutes, roll the dough out in a 14×8-inch (36×20-cm) rectangle. Spread the softened butter on top. Mix together the cinnamon and brown sugar. Sprinkle it all over the dough. Roll up the dough to make a 14-inch log. Cut into 10–12 even rolls and arrange in a lightly greased 9- or 10-inch round cake pan, pie dish, or square baking pan.
5Rise: Cover the pan with aluminum foil, plastic wrap, or a clean kitchen towel. Allow the rolls to rise in a relatively warm environment for 60–90 minutes or until double in size.
6Bake the rolls: After the rolls have doubled in size, preheat the oven to 375°F. Bake for 24–27 minutes, or until lightly browned. After 12 minutes of baking, loosely tent the pan with aluminum foil and continue baking to prevent heavy browning. If you want to be precise about their doneness, their internal temperature taken with an instant read thermometer should be around 195–200°F (91–93°C) when done. Remove pan from the oven and place pan on a wire rack as you make the frosting. (You can also make the frosting as the rolls bake.)
7Make the frosting: Beat the butter and cream cheese in a medium bowl with an electric mixer until blended. Add powdered sugar, half-and-half, and vanilla extract, and mix until smooth. Spread over warm rolls.
8
Cover leftover frosted or unfrosted rolls tightly and store in the refrigerator for up to 5 days.