Make the dough: In a medium bowl, whisk together the flour, sugar, salt, yeast, cinnamon and ginger. Set aside.
In a small saucepan, melt the butter over low heat. Add the milk and water and heat to 110°F. When the milk mixture comes to temperature, transfer to a large bowl or standing mixer. Mix in 1 egg on low speed.
Mix in the dry ingredients on low speed until the dough comes together, about 1 minute. Increase speed to medium-high and beat the dough for 4 minutes until the dough completely pulls away from the sides of the bowl. If you're not using a standing mixer, turn the dough out onto a lightly floured surface and knead for 4 minutes.
Once the dough is smooth and shiny, transfer it to a lightly greased bowl and cover it with plastic wrap or a kitchen towel and let rest for 10 minutes.
Shape the dough: Preheat the dough to 200°F. Turn the dough out onto a lightly floured surface. Shape into a rectangle, sprinkle the top lightly with flour and roll out to 14 x 8 piece.
Make the rolls: In a small bowl, combine the melted butter and molasses. Combine the sugar, cinnamon, and ginger in another bowl. Brush the melted butter and molasses over the dough then sprinkle with all of the sugar mixture. Using your fingers, break off pieces of the cream cheese and dot the dough.
Roll the dough from bottom to top using the longer side. Cut into 2 inch rolls and place in a lightly greased 8x8 pan or 9-inch round pan.
Cover with plastic wrap or a kitchen towel and place in the preheated oven. Shut the heat off and crack the oven door see notes. Let rise 45 to 60 minutes or until double in size. Remove from the pan from oven and preheat oven to 375°F.
Remove the plastic wrap or kitchen towel. Bake the rolls at 375°F for 18 minutes, cover with aluminum foil and bake for 7 to 10 minutes. Let cool for 5 minutes before frosting.
Make the frosting: In a small bowl, whisk together the cream cheese, confectioners sugar and vanilla. For a thinner frosting, add 1 to 2 tablespoons heavy cream. Once the frosting is mixed, spread over the rolls while they are still in the pan.