Combine water, yeast, and sugar. Cover and let sit for 5 minutes or until yeast looks foamy.
Add egg, canned pumpkin, and melted butter to the water and yeast mixture. Stir.
Add salt and flour (1 cup at a time) and mix until thoroughly combined.
Turn dough out onto a floured surface and knead until smooth (3-5 minutes). This dough will be a little stickier than normal bread dough. Just add enough flour to keep your hands from getting too sticky.
Add a tablespoon or two of oil to a large bowl (you can also spray it with nonstick cooking spray). Place dough in the bowl and turn the dough over once to coat in oil. Cover with a damp cloth or plastic wrap and let rise in a warm place for 1 hour.
After the dough has doubled in size, punch down and form into 24 rolls. Grease two 9x13 baking dishes (or a large baking sheet) and divide the rolls between the two pans.
Cover rolls and let rise for 30-40 minutes.
While rolls rise preheat the oven to 350 degrees.
Bake rolls for 18-20 minutes.
Brush generously with melted butter while hot.