Cook the butternut squash until tender, purée it until completely smooth, and measure 1 cup. Let the purée cool to lukewarm or room temperature before mixing it into the dough.

Stir the warm milk, yeast, and honey together in a large bowl. Let the mixture stand for 5-10 minutes, until foamy; if using instant yeast, it may be mixed directly with the flour.

Add the squash purée, egg, melted butter, and salt to the yeast mixture. Stir until the wet ingredients are evenly combined.

Mix in about 3 1/2 cups of flour, then add more flour 1 tablespoon at a time only if needed. Stop when the dough is soft and slightly tacky rather than stiff.

Knead the dough by mixer or by hand for 6-8 minutes, until smooth and elastic. Place it in a lightly greased bowl, cover, and let it rise in a warm place for 60-90 minutes, or until doubled.

Divide the dough into 12-16 equal pieces and shape each piece into a smooth ball. Arrange the balls in a greased 9-by-13-inch baking pan or on a lined baking sheet.

Cover the shaped rolls and let them rise for 30-45 minutes, until noticeably puffy. Near the end of this rise, heat the oven to 350°F (175°C).

Bake the rolls for approximately 18-22 minutes, until lightly golden and baked through in the centers. Brush them immediately with 1-2 tablespoons of melted butter and cool for several minutes before serving.