Combine brown sugar, ¼ cup white sugar, and cinnamon together in a small bowl; set aside.

Dissolve yeast and ½ teaspoon white sugar in warm water in the bowl of a stand mixer. Whisk together; let stand until yeast softens and begins to form a creamy foam, about 10 minutes.

Stir pumpkin purée, heavy cream, remaining ¼ cup white sugar, ¼ cup melted butter, salt, ½ teaspoon vanilla, ginger, and allspice into yeast mixer in the bowl of the stand mixer. Add 2 cups flour and egg; fit mixer with dough hook attachment and mix until combined, about 2 minutes.

Add remaining 2 ¼ cups flour; mix on low until dough is slightly sticky, smooth, and elastic, 6 to 7 minutes, adding more flour if mixture is too sticky.

Remove dough from hook; shape into a ball. Lightly coat the stand mixer bowl with vegetable oil, return dough to the bowl, turning to coat with oil. Cover bowl with aluminum foil; set aside to rise in a warm place until doubled in volume, about 1 ½ hours.

Generously butter a 13x9-inch baking dish.

Turn dough out onto a well-floured surface. Flatten into a rectangular shape about 1-inch thick using your hands. Generously sprinkle both sides of dough with flour; roll out to about a 20x12-inch rectangle. Pour 5 tablespoons melted butter over top; evenly brush over surface, leaving 2 inches along one wide edge unbuttered. Sprinkle brown sugar mixture over dough; moisten unbuttered edge with water. Roll dough tightly into a log, starting at the buttered wide edge; press firmly along moistened wide edge to seal.

Trim and discard uneven ends from the roll. Slice rolled dough into 16 equally-sized pinwheels; place pinwheels, cut-sides up, in the prepared baking dish. Cover with plastic wrap; set aside to rise in a warm place until doubled in volume, 45 to 60 minutes.

Preheat the oven to 350 degrees F (175 degrees C).

Bake in the preheated oven until golden brown, about 30 minutes; set aside to cool slightly while mixing glaze.