In a medium bowl, mix the wet ingredients: pumpkin puree, egg, milk, and melted butter.
In a large bowl or the bowl of your stand mixer, mix the dry ingredients: flour, sugar, instant yeast, salt, and pumpkin spice (if using).
Combine the wet ingredients with the dry ingredients, mixing until a shaggy dough forms. Knead the dough with a stand mixer (using the dough hook) for 8 minutes. If kneading by hand, it should take about 10-12 minutes.
The dough should be tacky and silky with a good windowpane at the end. Cover the dough and let it rise in a warm spot for the first rise, which should take about 1 hour and 30 minutes, with the dough rising to about 85% of its original size.
On colder days, the rising time might be a bit longer. You can use your oven’s proofing function or leave the dough in the oven with the light on.
Deflate the dough and divide it into 9, 12, or 16 equal parts, depending on the size you prefer.
Fold each piece of dough over itself 4 to 6 times, then pinch the seams together to close, forming a small ball of dough.
Place the balls seam-side down on a lined baking sheet or in an oven-safe dish. Let them rise for about 1 hour, or until they look puffy and light.
Just before baking, brush the top of the buns with the egg yolk.
Preheat your oven to 350°F and bake for 25-30 minutes.
Let the buns cool for 10 minutes before serving.