Combine 2 cups flour, sugar, dry yeast and salt in a large mixer bowl and stir until blended. Place water, pumpkin 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 of electric mixer, 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.
For filling, combine brown sugar and pumpkin pie spice in a small bowl. Set aside.
Roll dough into a 15 x 12-inch rectangle using a rolling pin. Spread 3 tablespoons butter over dough stopping at least 1/2-inch from the edges on the long sides. Sprinkle with sugar mixture. Beginning at long end of each rectangle, roll up tightly. Pinch seams to seal. Cut into 12 equal pieces. Place, cut sides down, in greased 13 x 9 inch baking pan. Cover with towel; let rise in warm place until doubled in size, about 45 minutes.
Bake in preheated 350°F oven for 25 to 30 minutes or until rolls are golden brown. Cool on wire rack for at least 20 minutes.
Combine all frosting ingredients in a large bowl and beat, adding just enough milk until frosting is creamy. Spread over rolls.