Mix the dough: In a stand mixer bowl, whisk together flour, yeast, brown sugar, and salt. In another bowl, mix the warm milk, pumpkin, spice, softened butter, and egg. Pour the wet ingredients into the dry ingredients.
Knead the dough: Using a stand mixer with a dough hook, knead for 5-7 minutes until the dough is smooth. If kneading by hand, do so on a floured surface for 10-12 minutes.
First rise: Place the dough in a greased bowl, cover it, and let it rise in a warm spot for 60-90 minutes, or until it has doubled in size.
Shape the rolls: Punch down the dough and divide it into 24 equal pieces. Roll each piece into a ball and place them in a greased 9x13-inch pan.
Second rise: Cover the pan and let the rolls rise for another 30 minutes. Meanwhile, preheat your oven to 375°F (190°C).
Bake and brush: Bake for 15-20 minutes until golden brown. As soon as they come out of the oven, brush the tops with the melted butter.