How to Make It
In a small saucepan, combine the milk, squash puree and sugar and cook over low heat, whisking, until the milk is lukewarm and the sugar is dissolved. Remove the pan from the heat. Whisk in the yeast and egg; let stand until foamy, 10 minutes.
In the bowl of a stand mixer fitted with the paddle, beat the 4 1/4 cups of flour and the salt at low speed just to combine. With the machine on, slowly add the warm milk mixture and beat until incorporated, about 1 minute. Beat in 1 tablespoon of the butter until completely incorporated; scrape down the side of the bowl. Add the remaining 7 tablespoons of butter, 1 tablespoon at a time. Beat the dough until well combined, about 2 minutes.
Fit the mixer with the dough hook. Knead the dough at medium speed until smooth, occasionally scraping down the side of the bowl and dusting with 1 to 2 tablespoons of flour if the dough sticks, about 5 minutes. Transfer the dough to a lightly oiled bowl, turn to coat with oil and cover with plastic wrap. Let stand in a warm place until doubled in bulk, about 2 hours.
Line 2 baking sheets with parchment paper. Gently press down on the dough; it should be tacky. Divide the dough into 36 equal pieces and cover with plastic wrap. Gently squeeze 1 piece of dough through your thumb and index finger to form a ball; pinch the bottom. Place the ball pinched side down on the prepared sheet. Repeat with the remaining pieces of dough, spacing the balls 1 1/2 inches apart. Loosely cover the rolls with plastic wrap and let rest in a warm place until doubled in size, about 1 hour.
Preheat the oven to 400°. Brush the tops of the rolls with the melted butter. Bake for 15 to 18 minutes, until golden and cooked through, shifting the pans from top to bottom and front to back halfway through baking. Serve warm.
To make the puree, boil 1 cup of 1-inch-diced butternut squash for 20 minutes, then drain and puree.