In a large bowl or stand mixer, combine warm water, sugar, and yeast. Let stand for 5–7 minutes until foamy and aromatic.
Whisk melted butter, pumpkin purée, warm milk, egg, minced rosemary, and fine sea salt into the bloomed yeast mixture until smooth.
Add 3 ½ cups of flour to the bowl. Using the dough hook (or a wooden spoon), knead on medium-low speed for 8–10 minutes until smooth, elastic, and slightly tacky. Add remaining flour 1 tablespoon at a time only if the dough is too wet to hold shape.
Transfer the dough to a lightly oiled bowl, cover with a damp cloth or plastic wrap, and let rise in a warm spot for 1 to 1 ½ hours, or until doubled in size.
Divide the dough into 12 equal portions and shape into rolls. Grease a baking dish and arrange the rolls inside.
Cover the shaped rolls loosely with plastic wrap or a clean towel and let them rise for another 35 to 45 minutes. While the rolls are rising, preheat your oven to 350°F.
Bake the rolls for 22 to 26 minutes, or until the tops are deep golden brown.
While the rolls are still hot, combine the melted butter with the finely chopped fresh rosemary. Brush the rosemary butter generously over the tops of the rolls. Finish with a generous sprinkle of flaky sea salt.
