Making the Dough:
In the bowl of a stand mixer fitted with a paddle attachment, combine 3 cups of flour, yeast, brown sugar, salt, and ground ginger.
Add softened butter and warm milk to the mixture and blend on medium speed until combined.
Add eggs, molasses, and vanilla extract, mixing until well blended.
Switch to the dough hook attachment. Add the remaining 2 cups of flour and mix on low/medium speed, ensuring the flour is fully incorporated. Scrape down the sides of the bowl as needed.
Turn the mixer to high speed and knead the dough for 4-5 minutes until smooth. If the dough is too sticky, add 1 tablespoon of flour and continue kneading. The dough should not stick to your fingers when pressed lightly.
Cover the dough in the bowl with plastic wrap and allow it to rest at room temperature for 20 minutes while you prepare the filling.
Making the Filling:
In a small bowl, mix together the brown sugar, cinnamon, ginger, butter, and molasses until well combined.
Grease a 9×13-inch baking dish with butter or cooking spray.
On a lightly floured surface, roll the dough into a 12×16-inch rectangle. Spread the filling evenly over the dough, leaving the edges untouched.
Starting with the long side, roll the dough into a log, keeping it tight as you roll. Pinch the edges to seal the log. Cut into 12 equal rolls using a sharp knife.
Place the rolls in the prepared baking dish, cover with plastic wrap, and let rise at room temperature for 1 hour.
Baking the Rolls:
Preheat the oven to 350°F (175°C). Bake the rolls for about 40 minutes, or until the edges are golden brown and the center is puffed.
Making the Frosting:
In a medium bowl, beat together the softened butter and cream cheese until smooth.
Add molasses, ginger, cinnamon, and vanilla extract, mixing until combined.
Gradually add powdered sugar, beating on high speed until the frosting is smooth and creamy.
Finishing the Rolls:
Spread the gingerbread cream cheese frosting over the warm cinnamon rolls.
Serve warm and enjoy!