In a large mixing bowl, combine the warm milk, yeast, and a teaspoon of the sugar. Let it sit for 5 to 10 minutes until foamy.
Add the pumpkin puree, melted butter, egg, vanilla, remaining sugar, and salt. Mix well.
Gradually add the flour, mixing until a soft dough forms.
Knead the dough for 8 to 10 minutes until smooth and elastic.
Place the dough in a lightly greased bowl, cover, and let it rise for about 1 hour or until doubled in size.
Punch down the dough and roll it into a rectangle about 12x18 inches.
Spread the softened butter evenly over the dough.
Mix the brown sugar, cinnamon, and pumpkin pie spice together, then sprinkle evenly over the butter.
Roll the dough tightly into a log, starting from the long side.
Cut into 12 equal rolls and arrange them in a greased 9x13-inch baking dish.
Cover and let the rolls rise for another 30 to 45 minutes.
Preheat the oven to 350°F (175°C).
Bake for 25 to 30 minutes, or until the rolls are golden brown and cooked through.
Meanwhile, beat together the cream cheese, butter, powdered sugar, and vanilla until smooth and fluffy.
Spread the frosting over the warm cinnamon rolls before serving.