Combine 2 cups flour, sugar, dry yeast and salt in a large mixer bowl and stir until blended. Place water and butter in a microwave-safe bowl. Microwave on HIGH in 15 second increments until very warm but not hot to the touch (120° to 130°F. Butter won’t melt completely). Add to flour mixture with egg.
Beat 2 minutes at medium speed, scraping bowl occasionally. Add 1 cup flour; beat 2 minutes at high speed, scraping bowl occasionally. Stir in just enough remaining flour so that the dough will form into a ball.
Knead on lightly floured surface until smooth and elastic and dough springs back when lightly pressed with 2 fingers, about 6 to 8 minutes. Cover with a towel; let rest for 10 minutes.
While dough rests, prepare the filling: Combine sugar, pumpkin pie spice, and pumpkin puree in a small bowl. Set aside.
Roll dough into a 15 x 10-inch rectangle using a rolling pin. Spread the filling evenly over the dough, stopping at least 1/2-inch from the edges on the long sides.
Beginning at the long end of the rectangle, roll up tightly. Pinch seams to seal. Cut into 12 equal pieces.
TIP: Use unflavored dental floss instead of a knife to cut the rolls. To do this, cut a piece of floss about 12 inches long. Slide floss under the roll; bring the ends up and cross over to cut each slice.
Place, cut sides down, in a greased 13 x 9 inch baking pan.
To bake immediately:
• Cover with a towel; let rise in a warm place until doubled in size, about 1 hour .
• Bake in a preheated 350ºF oven for 25 to 30 minutes or until rolls are golden brown. Cool on a wire rack for at least 20 minutes.
To bake the next day:
• Cover the pan with plastic wrap and refrigerate overnight.
• The next morning, remove pan from the refrigerator and let sit at room temperature for 30-45 minutes. Bake in preheated 350ºF oven for 25 to 30 minutes or until rolls are golden brown. Cool on a wire rack for 10-15 minutes before frosting.
For the frosting:
Beat cream cheese, butter and vanilla in a large bowl. Slowly add 1 cup of sifted powdered sugar at a time, beating well after each addition. Add milk ½ tablespoon at a time until frosting is desired consistency. Frost rolls.