In a large bowl or the bowl of a stand mixer fitted with the kneading hook, stir together the yeast, salt, nutmeg and 3 cups of the flour.
In a medium saucepan, combine the milk, butter and honey. Heat over medium heat, stirring, until butter is melted and mixture is about 120-130 F. (If it gets too warm, simply stir for a bit until temperature reduces in to this range). Add the milk mixture to the dry ingredients and mix at medium speed for about 2 minutes. Add pumpkin puree and mix for another 2 minutes. Begin adding additional flour, 1/2 cup at a time, to start, and then less as you progress, until you have a moist, smooth but not sticky dough. Turn dough out on to a floured surface and knead a few minutes. Move to a greased bowl, cover with plastic wrap and allow to rise until doubled, about 45 minutes to 1 hour.
Meanwhile, grease two 9-inch round cake pans and set aside.
*Note: I like to use a kitchen scale to ensure my rolls all come out the same size. It ensures even cooking and it looks nicer in the pan, as well. Simply weigh out the whole piece of dough. Divide that weight by 2 to determine the weight each half should be. To divide each half in to 12 equal pieces, divide the weight of the half by 12 and make all the rolls that weight.
One dough has doubled, remove dough to a floured surface and gently deflate. Divide dough in to 2 equal parts, then divide each part in to 12 equal size pieces, for a total of 24 pieces. Form each of the pieces in to a ball. Place 12 rolls in to each pan, with 9 around the outside and 3 in the middle. Cover with a clean tea towel and set aside to rise until doubled, about 45 minutes.
Preheat oven to 375F. about 15 minutes before rising is complete. Once rolls have doubled, brush the tops and sides of the dough with the beaten egg, then sprinkle tops with seeds. Doing 4 of each of sesame seed, poppy seed and pumpkin seeds is nice.
Place in preheated oven and bake for 22-25 minutes, or until cooked through and lightly golden brown. *Check at the 20 minute mark and loosely cover with aluminum foil if rolls are browned enough at that point. Allow to cool in the pan for 5 minutes, then remove as one piece from the pans to cooling racks to cool completely. Avoid tearing off rolls before the rolls have almost completely cooled to prevent gummy dough.
These rolls freeze well, so can be made ahead. Wrap tightly and freeze. Thaw in the same wrapping. Re-heat in a 350F. oven wrapped in foil until warmed, about 8-10 minutes.Be sure to read the “Ingredient and Cook's Notes" (above the recipe card!), where I share more detailedtips, variations and substitution suggestions for this recipe!