Pour milk into a large glass measuring cup. Add butter and microwave on high, until the mixture is warmed to the temperature directed by the yeast manufacturer (usually between 110 to 125 degrees F). Start with 30 seconds, then microwave in 10-second intervals, until the desired temperature is reached. The butter may not melt completely and the milk should feel warm but not hot.
Place the white whole wheat flour, brown sugar, egg, molasses, vanilla, yeast, salt, cinnamon, ginger, cloves, and nutmeg in the bowl of a stand mixer or a large mixing bowl. Pour in the heated milk and butter. Beat on low speed for 3 minutes, stopping occasionally to scrape down the sides of the bowl.
Add 2 1/2 cups all purpose flour. Beat on low speed, until the flour is absorbed and the dough is no longer sticky, scraping down the sides of the bowl as you go. If the dough is extremely sticky, add the remaining 1/2 cup flour one tablespoon at a time, combining after each, just until the dough begins to form a shaggy ball and pulls away from the sides of the bowl. It should still feel fairly tacky but not sticky.
If using a stand mixer, fit the mixer with a dough hook and knead on medium-low for 6 minutes, until smooth and elastic. If kneading by hand, turn the dough out onto a floured work surface and knead until smooth and elastic, about 8 minutes. Add a little flour, one tablespoon at a time, if the dough is too sticky and clings readily to your hands.
Lightly oil a large, clean bowl with cooking spray. Form dough into a ball and place it in the bowl, turning once to coat with oil. Cover bowl with plastic wrap or a clean kitchen towel. Let rise in a warm, draft-free place until doubled in volume, about 2 hours.
Meanwhile, make the filling. Stir together brown sugar, granulated sugar, cinnamon, ginger, and cloves. Set aside. Line a 9x13-inch pan with parchment paper and lightly coat with cooking spray.
Once the dough has risen, turn out onto a lightly floured surface and roll into a 12x20-inch rectangle. Brush all over with 3 tablespoons melted butter and sprinkle all of the sugar and spice filling over the top.
Starting at one of the short edges, roll the dough into a tight log, then tightly pinch the seam to seal. Gently stretch and pat the dough log as needed so that it is roughly even in diameter and 18 inches in length. With a serrated knife, gently saw the log into 12 segments (each segment will be 1 1/2 inches wide).
Arrange the rolls in the prepared pan, cut sides up. Cover the baking dish with plastic wrap or a clean kitchen towel and let rise in a warm, draft-free place until doubled in volume, about 45 minutes. Alternatively, you can place the rolls in the refrigerator and let them rise overnight. Let the rolls come to room temperature for 30 minutes before baking.
Place rack in center of your oven and preheat to 350 degrees F. Bake the rolls until golden and the internal temperature reaches 185 degrees F, about 25 to 35 minutes. Check the rolls at the 20-minute mark. If browning too quickly, gently tent with foil and continue baking.
For the frosting, beat together the cream cheese and powdered sugar until smooth in a medium bowl. Add 3 tablespoons milk, vanilla, cinnamon, and cloves. Add a little additional milk as needed to reach the desired consistency. Frost the rolls and serve warm.
Notes
TOSTORE: Baked gingerbread cinnamon rolls may be stored, covered in an airtight container, for up to 3 days in the refrigerator.
TO FREEZE: Baked gingerbread cinnamon rolls may be stored in a freezer for up to 3 months. For best results, store unfrosted and wrapped tightly in plastic with an outer layer of foil. Defrost overnight in the fridge before reheating and then topping with frosting.
TO REHEAT: Warm cinnamon rolls in an oven preheated to 350 degrees F until heated through. Alternatively, you may also heat individual rolls in the microwave for a few seconds until warm.
TO MAKE AHEAD: These rolls are perfect to make ahead and let rise overnight in the refrigerator. Once the rolls are shaped, arrange them in the pan, cover with plastic wrap, then let rise in the refrigerator overnight. The next morning, set the pan out at room temperature for 30 minutes, then bake as directed.