Combine milk, brown sugar, 1 tablespoon butter, and salt in a small saucepan over medium heat; heat to about 110 degrees F (43 degrees C), 2 to 4 minutes. Remove from heat.
Mix warm water and yeast together in a bowl until yeast is dissolved. Combine yeast mixture and milk mixture in a large bowl. Add pumpkin puree, all-purpose flour, whole wheat flour, cinnamon, ginger, and nutmeg; beat until dough is smooth, adding more all-purpose flour if needed to form a soft dough.
Grease a large bowl.
Turn dough onto a floured surface and knead until smooth and elastic. Place dough into the greased bowl, turning once to grease the top of the dough. Cover and let rise in a warm place until doubled in size, about 1 hour.
Grease an 9x13-inch baking dish.
Punch dough down and divide into 10 pieces. Shape each dough piece into a ball. Place dough balls into the prepared baking dish; cover and let rise until doubled in size, about 30 minutes.
Preheat oven to 375 degrees F (190 degrees C).
Melt remaining butter in a saucepan over medium-low heat, 2 to 3 minutes. Brush melted butter over dough.
Bake in the preheated oven until golden brown, 20 to 25 minutes. Transfer rolls to a wire rack to cool slightly before serving.