In large bowl of electric mixer fitted with dough hook, combine warm milk with yeast. Mix until yeast is dissolved. Let stand 5 minutes. Add egg, squash puree, sugar and salt; mix until well blended. Add flour; mix until a soft dough forms, about 4 to 6 minutes. Dough should be moist but not soft and sticky. If necessary, add additional flour 2 tablespoons at a time until a smooth dough forms. Add softened butter until well-blended. Transfer dough to a medium bowl. Cover with plastic wrap and allow dough to rise in a warm, draft free place until doubled in size, 45 minutes to 1 hour.
Turn dough out onto a lightly floured surface; divide evenly into 12 pieces (about 2 ounces each). Roll each piece into a ball. (For for a more traditional shape, take a small wooden dowel, such as the handle of a wooden spoon, and press straight down on the top of the roll just until 2 clefts form in the roll.) Space rolls evenly in an 8- or 9-inch square baking pan. Cover and allow to double in size, 45 minutes to 1 hour.
Meanwhile, heat oven to 350 degrees F. When rolls are ready to bake, lightly brush tops with melted butter and sprinkle with coarse salt. Bake until golden brown, 16 to 18 minutes.
Serve warm or at room temperature.