Warm the milk to approximately 110°F either in the microwave or on the stove, it should feel warm, but not hot. Add it to the bowl of a stand mixer along with the yeast and sugar. Let sit for about 10 minutes until nice and frothy.
Add the 1/2 cup pumpkin puree, egg and salt to the milk mixture and turn the mixer on low just to combine the ingredients.
Add the flour and mix until it comes together to form a dough, then cut the softened butter into small cubes and add to the dough. When you first add the butter the dough will become very sticky, but just let the mixer do its job and it will all come together.
Turn the mixer up to medium speed and allow it to mix for 8-10 minutes until all the dough has pulled away from the bowl and you have a soft, elastic dough.
Once the dough is soft and elastic, turn off the mixer and turn the dough onto the counter. Knead a few times and form a ball, then return the dough to the bowl, cover and let rise until doubled. This will take 1-2 hours depending on the warmth and humidity of your home.
After the dough has risen, turn it out onto a floured counter or pastry mat and roll it into a rectangle of about 15x12 inches with the long end facing you.
Mix the 3/4 cup pumpkin puree, melted butter, brown sugar and pumpkin pie spice in a small bowl.
Spread the pumpkin mixture over the dough leaving 1 inch on the long end furthest from you bare.
From the long end closest to you, roll the dough to form a log, then cut into 12 rolls. Place the rolls into a greased 9x13 baking pan, then cover and let rise for approximately 30 minutes.
While the rolls are rising, preheat the oven to 375°F
Once the rolls have risen, uncover and bake on the middle rack for 25-30 minutes until golden brown.
Remove from the oven, let cool for 5-10 minutes and ice as desired.