Bloom Yeast: In the bowl of a stand mixer fitted with the dough hook, whisk warm whole milk, yeast, and 1 teaspoon of brown sugar. Let sit for 5 to 8 minutes until foamy and fragrant.
Knead Spiced Dough: Add molasses, remaining brown sugar, eggs, melted butter, fine salt, ground ginger, cinnamon, nutmeg, cloves, and 3 1/2 cups flour. Knead on low speed for 8 to 10 minutes, gradually adding remaining flour if dough feels too sticky, until a silky, smooth, slightly tacky dough forms.
First Rise: Shape dough into a smooth ball. Place in a lightly greased bowl, cover with a damp kitchen towel or plastic wrap, and let rise in a warm, draft-free spot for 60 to 75 minutes until doubled in bulk.
Roll & Fill: Turn dough onto a lightly floured surface. Roll into a large 12x18-inch rectangle. In a small bowl, whisk softened butter, brown sugar, molasses, cinnamon, and ginger into a spreadable paste. Spread evenly across the dough with an offset spatula, leaving a 1/2-inch border along the top edge.
Slice & Second Rise: Tightly roll dough up from the long edge into an 18-inch cylinder. Using baker's twine, unflavored dental floss, or a serrated knife, slice into 12 equal rolls. Place rolls in a greased 9x13-inch metal baking pan. Cover and let rise for 35 to 45 minutes until puffed and touching.
Bake to Golden Perfection: Preheat oven to 350°F (175°C). Pour 1/3 cup warm heavy cream over the tops of the rolls right before baking (the secret to ultra-tender bakery chew!). Bake for 22 to 26 minutes until tops are lightly golden brown and centers are set.
Glaze & Serve: While rolls bake, beat softened cream cheese, butter, powdered sugar, vanilla, and heavy cream until silky smooth. Spread half of the icing over the rolls immediately upon removing from the oven so it melts into the warm crevices, then slather the remaining icing after 10 minutes. Serve warm!