In the bowl of a stand mixer, combine warm milk and yeast. Let sit for 5-10 minutes until foamy and bubbly. Add pumpkin puree, sugar, melted butter, and egg. Mix on low speed until combined. In a separate bowl, whisk together 3½ cups flour, salt, pumpkin pie spice, and cinnamon. With mixer on low, gradually add flour mixture until a soft dough forms. Switch to dough hook and knead for 5-7 minutes, adding more flour 1 tbsp at a time if dough is too sticky.

Place dough in a greased bowl, turning to coat all sides. Cover with plastic wrap or a damp kitchen towel. Let rise in a warm, draft-free spot for 1-1½ hours, or until doubled in size. The dough is ready when you poke it and the indentation remains.

Punch down risen dough and turn out onto a lightly floured surface. Roll into a 16x12 inch rectangle. In a small bowl, mix together softened butter, brown sugar, flour, pumpkin pie spice, cinnamon, pumpkin puree, and salt until it forms a spreadable paste. Spread filling evenly over dough, leaving a ½-inch border on one long edge. Starting from the long edge opposite the border, tightly roll dough into a log. Pinch seam to seal. Cut log into 12 equal rolls (about 1½ inches each) using a sharp knife or dental floss.

Place rolls in a greased 9x13 inch baking pan, cut side up, leaving space between each. Cover and let rise for 30-45 minutes until puffy and nearly doubled. Preheat oven to 350°F during last 15 minutes of rising. Bake for 22-27 minutes until tops are golden brown and centers are cooked through (190°F internal temperature). Let cool in pan for 10 minutes, then transfer to wire rack. Frost while still warm.