Combine milk, 1 teaspoon of brown sugar, and yeast in a bowl. Set aside for 5 minutes until foamy.
In a large bowl, beat remaining brown sugar, eggs, molasses, butter, ginger, cinnamon, nutmeg, cloves, and 1 cup of flour until combined. Add yeast and milk mixture once it is foamy and mix well.
Continue to add flour, 1 cup at a time, switching to the dough hook and kneading for 5-7 minutes to form a soft, smooth dough. Add remaining ½ cup of flour if needed.
Scrape dough into a clean, oiled bowl and cover with plastic wrap. Let rise for 1 hour in a warm spot.
In a medium bowl, combine brown sugar, granulated sugar, molasses, cinnamon, ginger, nutmeg, cloves, and softened butter. Mix until combined. Set aside.
Turn risen dough out onto a generously floured surface. Roll or press out dough with a floured rolling pin or floured hands to prevent sticking to form a large 10x16-inch rectangle.
Spread filling evenly over dough. Roll from the long edge into a tight log.
Use a sharp knife, a thin string, or piece of unflavored dental floss, cut dough into 12 evenly-sized rolls. Transfer to a greased 9x13-inch baking pan. Cover with plastic wrap and let rise for 20-30 minutes. Preheat the oven to 350°F.
Bake for 25-30 minutes, until golden brown. Cool for 5-10 minutes before frosting.
Make frosting by beating butter and cream cheese in a medium bowl with a hand mixer until creamy and smooth. Add powdered sugar, vanilla, and cinnamon. Beat again. Add enough heavy cream to get desired consistency.
Spread frosting over slightly cooled rolls. Serve warm.