Combine 2 cups flour, sugar, dry 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 hot to the touch (120° to 130°F. Butter won’t melt completely). Add to flour mixture with egg.
Beat 2 minutes at medium speed with an electric mixer, scraping bowl occasionally. Add 1 cup flour; beat 2 minutes at high speed, scraping bowl occasionally. Stir in just enough remaining flour so that the dough will form into a ball.
Knead on lightly floured surface until smooth and elastic and dough springs back when lightly pressed with 2 fingers, about 6 to 8 minutes. Cover with a towel; let rest for 10 minutes.
For the filling, combine the brown sugar, molasses, spices, and butter in a microwaves safe bowl. Microwave in 30 second increments until the sugar is melted and the mixture is smooth.
Roll dough into a 15 x 10-inch rectangle using a rolling pin. Spread the filling over the dough, right to the edges. Beginning at long end of the rectangle, roll up tightly. Pinch seams to seal. Cut into 12 equal pieces using a very sharp knife or unflavored dental floss instead of a knife to cut rolls. If you do use floss, slide a long piece of floss under the roll; bring the ends up and cross over to cut each slice. Place, cut sides down, in greased 13 x 9 inch baking pan . Cover with towel; let rise in warm place until doubled in size, about 1 hour .
Bake in preheated 350ºF oven for 25 to 30 minutes or until rolls are golden brown. Cool on wire rack for at least 20 minutes.
Combine all frosting ingredients in a large bowl and mix together until light and fluffy and no streaks of powdered sugar remain. Frost the gingerbread cinnamon rolls while they’re still warm. Serve hot!