Combine 2 cups flour, sugar, yeast, and salt in a large mixer bowl and stir until blended. Place water and butter in a microwave-safe bowl. Microwave on HIGH in 15 second increments until very warm but not too hot to touch. (Butter won’t be completely melted) Add to flour mixture, then mix in the egg.
Mix the dough 2 minutes at medium speed, scraping down the bowl as needed. Add 1 cup flour, mix 2 minutes at high speed, then stir in just enough remaining flour so that the dough will form into a ball.
Knead on lightly floured surface until smooth and elastic, about 6-8 minutes. Cover with a towel and let rest for 10 minutes.
To make the filling, stir together sugar, brown sugar, and cinnamon, set aside.
Roll the dough into a 15×10-inch rectangle. Spread the softened butter over the dough, leaving a 1/2-inch gap from the edges. Sprinkle cinnamon and sugar mixture over buttered dough.
Beginning at the long end of the rectangle, tightly roll the dough into a log, pinching the seams to seal. Cut into 12 equal pieces, place pieces cut side down in a greased 9×13 inch pan. Cover with a towel and place in a warm place to rise for 1 hour.
Heat oven to 350 degrees. Bake uncovered for 25-30 minutes until golden brown. Allow to cool 15 minutes before frosting.
To make the frosting, melt butter in a medium sauce pan and boil for 2 minutes. Stir in brown sugar, reduce heat to medium-low and boil for 2 minutes. Stir in milk, remove from heat and pour into a large bowl. Stir in powdered sugar. Add milk and powdered sugar, whipping with an electric mixer medium-high speed, until light and fluffy. Spread on cinnamon rolls.