In a small saucepan, melt butter. When butter is almost melted, turn off the heat. When the butter is fully melted add the milk. Let the mixture cool a few minutes until it is about 110 degrees F.
Add the sugar and the yeast, mix to combine, and lest rest a few minutes until the yeast becomes bubbly.
In the bowl of a standing mixer, using the paddle attachment, combine the yeast mixture with the pumpkin, eggs, salt, and one half of the flour. Mix on medium speed for about 3 minutes.
Remove the paddle attachment and put on the dough hook. Add most of the remaining flour, reserving about ½ cup, and knead at a medium speed, adding more flour as necessary, until the dough is smooth and does not stick to the side of the bowl, about 10 minutes.
Shape the dough into a ball and put in a large, oiled bowl. Cover with plastic wrap and let rise in a warm, draft free place until doubled in size. This should take about 1 ½ to 2 hours depending on the temperature of your dough and the temperature of your kitchen,
After the dough has risen and doubled in size, punch it down and place it on the counter. I don't usually need to flour the work surface, but if you have any problems with the dough sticking you can use a light dusting of flour.
Roll the dough out into a 16 by 12 inch rectangle with the long side facing you.
Mix the brown sugar, cinnamon, and salt in a small bowl.
Spread the pumpkin puree over the rolled out dough, leaving a ½ inch border at the far edge for sealing the dough.
Sprinkle the cinnamon sugar mixture evenly over the pumpkin puree.
Roll up the dough, jelly roll style, starting at the long edge closest to you, making a tight, uniform cylinder.
Pinch the ½ inch border to the cylinder to seal the roll.
Using your hands, press gently on the ends of the cylinder to ensure a uniform size and shape.
Using a serrated knife and a sawing motion to prevent squishing the roll, gently cut the roll into 12 equal slices. I like to cut the cylinder in half, cut the halves in half, then cut each fourth into thirds.
Put a piece of parchment or baking paper onto the bottom of a 9 by 13 inch baking pan. I like to use clear glass to see the color of the bottom of the rolls while baking.
Place the rolls, cut side up, evenly into the baking dish.
Cover the dish with plastic and set in a warm, draft free spot to rise until doubled, about 1 ½ to 2 hours.
When the rolls are almost fully risen, preheat the oven to 350 degrees F.
When the rolls are fully risen, bake in the middle of the preheated oven until they are golden brown, about 30 minutes.
Let the rolls cool for about 5 minutes, then invert the rolls onto a cooling rack and let cool for about 10 minutes before inverting them back onto a serving dish.