Combine your flour, sugar, and salt in a large bowl (stand mixer bowl if you have one) and set aside.
Melt butter in the microwave, and then add in the milk. You need this mixture to be warm, 100-105F (do not let it go above 110F). Whisk in your yeast until it has dissolved and then pour mixture into your dry ingredients and add egg. You can either stir this together by hand, or use the paddle attachment on your stand mixture until a soft dough forms.
Remove from bowl and knead on a lightly floured surface for 3-5 minutes until your dough is a smooth ball. Place your dough ball into a lightly oiled bowl and cover with a tea towel to rest for about 10 minutes while you get the filling ingredients ready.
Roll your dough out on a lightly floured surface into a 14" x 8" rectangle. Once rolled out, spread softened butter all over the surface and sprinkle with your brown sugar and cinnamon. Roll up the dough into a long log, and cut into 12 rolls. Transfer your rolls to a greased 9" round cake pan or pie dish. Cover your rolls and allow them to rise for 60-90 minutes, or until they double in size.
Once rolls have doubled, bake them in a 375F oven for 22-25 minutes, or until they're lightly browned. I test them with a thermometer, they ought to come out 195-200F. Remove pan and place on a wire rack.
While the rolls cool, make your icing. Using a stand mixer (or handheld mixer, or a whisk) beat the cream cheese until smooth are creamy, add the butter until smooth and combined, then beat in the confectioners' sugar and vanilla until combined. Spread over your rolls and serve!
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