Heat the milk to 95–100°F. In a medium bowl, combine the yeast, warm milk, and sugar. Mix with your hands until combined, cover with plastic wrap, and set in a warm spot until visibly bubbly, about 20 minutes.

In the bowl of a stand mixer, add the remaining flour, dark brown sugar, salt, softened butter, pumpkin puree, spice mix, egg, and the sponge. Mix on low speed with the paddle attachment until combined.

Switch to the dough hook and mix on medium speed until the dough passes a rough window pane test: pinch off a piece, gently pull and rotate into a square shape. The dough should not tear, and you should be able to see through portions of it without it ripping.

Scrape the dough into a well-oiled bowl and cover with plastic wrap. Allow to proof for 30 minutes in a warm place, then fold: grab one side, pull it upward, and fold it over one-third of the dough. Rotate the bowl 180 degrees and repeat. Rotate a quarter turn and repeat twice more until you have a tight square packet.

Cover again and allow the dough to double, about an additional 30 minutes. Meanwhile, grease a 9 x 13-inch baking dish with cooking spray or butter.

Turn the doubled dough out onto an un-floured surface and divide into 12 equal portions, approximately 99g each. Place equidistant in the prepared dish, cover with plastic wrap, and proof until doubled, about 30 minutes.

Preheat the oven to 350°F. When the rolls have doubled and feel light and airy, brush with beaten egg.

Bake for 20 to 22 minutes, until an instant-read thermometer inserted into a center roll reads 185°F. If you don't have a thermometer, a toothpick should come out with clinging crumbs and no streaks of raw dough.

Immediately brush the hot rolls with melted butter and serve warm.