Warm the milk to 100–110°F (38–43°C). Combine it with the brown sugar and active dry yeast in the bowl of a stand mixer. Let stand for 5–10 minutes, or until foamy.
Add the flour, butter, two of the eggs, pumpkin puree, Italian cheese blend, rubbed sage, salt, and black pepper. Mix on low speed just until a shaggy dough forms.
Switch to the dough hook and knead on medium speed for 8 minutes, until the dough is smooth and elastic.
Cover and let the dough rise in a warm place until doubled in size, about 1½ hours.
Divide the dough into 20 equal portions. Shape each into a smooth ball by pulling the edges underneath and pinching the seam closed.
Place the rolls seam side down on a silicone-lined baking sheet. Cover and let rise until puffy, about 30 minutes.
Preheat the oven to 375°F (190°C).
Beat the remaining egg with 1 teaspoon water and brush the tops of the rolls. Score each roll with a shallow slash.
Bake for 15 minutes, watching closely during the last few minutes as the tops can brown quickly.
Transfer the rolls to a wire rack and cool before serving.
Notes
Warm the milk to 100–110°F (38–43°C) to properly activate the active dry yeast.
The dough is fully risen when it has doubled in volume, not simply after the recommended rise time.
If the tops begin browning too quickly, loosely tent the rolls with aluminum foil for the remaining baking time.
Store cooled rolls in an airtight container at room temperature for up to 3 days or refrigerate for up to 1 week.
Freeze cooled rolls for up to 3 months. Freeze them in a single layer until firm, then transfer to a freezer-safe bag or airtight container to prevent them from sticking together. Reheat in a 325°F (165°C) oven until warmed through.