In a large mixing bowl, combine all-purpose flour, brown sugar, active dry yeast, ground cinnamon, and ground ginger. In a separate bowl, heat milk until warm (about 110°F/43°C) and mix in melted butter. Slowly pour the warm milk mixture into the dry ingredients, followed by the egg. Stir until combined, then knead the dough on a floured surface for about 5-7 minutes until smooth and elastic.

Place the kneaded dough in a greased bowl, turning it to coat, and cover it with a clean kitchen towel. Let it rise in a warm environment for about 1 hour or until it has doubled in size.

Once the dough has risen, punch it down gently to remove excess air. On a floured surface, roll the dough into a rectangle about 1/4 inch thick. Spread softened butter evenly across the surface, then sprinkle generously with a mixture of brown sugar, ground cinnamon, and a touch of ground ginger.

Starting from one long edge, carefully roll the dough tightly into a log. Cut the log into 12 equal pieces and place them cut-side up in a greased baking dish.

Cover the dish with a kitchen towel, allowing them to rise a second time for about 30-45 minutes.

Preheat your oven to 350°F (175°C) and bake the risen rolls for 25-30 minutes, or until they are golden brown on top.

Once baked, remove the rolls from the oven and let them cool for about 10 minutes before frosting. Prepare your cream cheese frosting by beating together softened cream cheese, powdered sugar, and vanilla extract until smooth. Spread the frosting generously over the warm rolls.