In a large bowl, combine the warm milk, yeast, and 1 teaspoon of granulated sugar. Let it sit for 5-10 minutes until foamy.
Stir in the remaining granulated sugar, salt, pumpkin puree, melted butter, and beaten egg into the yeast mixture. Mix until well combined.
Gradually add the flour, mixing with a spatula until a shaggy dough forms. Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until smooth and elastic. If the dough is too sticky, add a little more flour, one tablespoon at a time. Shape into a ball.
Place the dough in a lightly greased bowl, turning to coat. Cover with plastic wrap or a clean kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
While the dough rises, prepare the filling. In a small bowl, combine the softened butter, brown sugar, cinnamon, and pumpkin pie spice. Mix until well combined to form a paste.
Once the dough has risen, punch it down gently. Turn it out onto a lightly floured surface and roll it into a large rectangle (approximately 12×18 inches).
Spread the filling evenly over the dough, leaving a small border along one of the long edges.
Starting from the long edge with the filling all the way to the edge, tightly roll up the dough into a log.
Cut the log into 12 equal-sized rolls. You can use a sharp knife or unflavored dental floss for cleaner cuts.
Place the cinnamon rolls into a greased 8×8 inch or 9×13 inch baking dish, cut-side up. If using parchment paper, line the dish first.
Cover the baking dish loosely with plastic wrap or a kitchen towel and let the rolls rise for another 30 minutes, or until puffy.
Preheat your oven to 350°F (175°C) during the last 15 minutes of the second rise.
Bake for 20-25 minutes, or until golden brown on top and cooked through.
While the rolls are baking, prepare the glaze. In a medium bowl, beat together the softened cream cheese and softened butter until smooth. Gradually beat in the powdered sugar until combined. Stir in the vanilla extract and 2 tablespoons of milk. Add more milk, one tablespoon at a time, until the glaze reaches your desired drizzling consistency.
Once the cinnamon rolls are out of the oven, let them cool in the baking dish for about 10 minutes before drizzling generously with the glaze. Serve warm.