Start the dough: Combine ⅓ cup milk, 2 tablespoons butter, and granulated sugar in a small saucepan over medium heat, and heat until a thermometer registers between 125°F and 130°F. (If it goes above 130°F, let it cool to that range. Otherwise, you'll kill the yeast). Whisk in the yeast, and let it stand until foamy (about 10 minutes).
Beat the pumpkin, ¼ cup brown sugar, ½ teaspoon pumpkin pie spice, ½ teaspoon cinnamon, and ¼ teaspoon salt at medium speed with a heavy-duty stand mixer until combined. Add the milk mixture and egg, and beat until combined. Gradually add in 2 ¾ cups flour, beating after each addition until the mixture forms a dough, scraping down the sides of the bowl if needed. (The dough should still be a little sticky.)
Turn the dough out onto a lightly floured surface and knead until elastic and smooth, adding ¼ to ½ cup more flour if necessary. (The dough should no longer stick to your hands or the surface.)
Place the dough in a lightly greased bowl, and cover with plastic wrap. Let the dough rise in a warm place, free from drafts, 1 ½ hours or until doubled in size.
Punch the dough down and roll out into an 18- x 12-inch rectangle. Combine the melted butter, ½ cup brown sugar, 1 ½ tablespoons pumpkin pie spice, and ½ teaspoon cinnamon in a bowl. Spread the pumpkin spice cinnamon mixture evenly over the dough.
Roll up the dough tightly from the long side, and trim the edges. Cut the dough into 12 equal pieces, and place in a lightly greased 13- by 9-inch baking pan. Cover with plastic wrap. Let it rise in a warm place, free from drafts, 1 more hour or until doubled in size.
Preheat the oven to 325°F. Bake the cinnamon rolls 24 to 26 minutes or until done. Remove to a wire rack.
While the rolls bake, make the maple cream cheese frosting: Beat the cream cheese at medium speed with an electric mixer until creamy. Gradually add the powdered sugar, and beat until combined. Add the maple extract and milk, and beat until combined.
To serve, spread the maple cream cheese frosting over the warm cinnamon rolls.