Combine the warm water, yeast, sugar, butter, egg, salt, and flour in your stand mixer. Let the dough hook do the work for 5 to 10 minutes until the dough turns smooth, elastic, and pulls cleanly away from the bowl's sides.
Shape the dough into a ball and place it in a lightly greased bowl. Cover it and let it sit somewhere warm for 60 to 90 minutes, until it has doubled in size. This first rise builds the fluffy texture you want.
Punch down the risen dough and turn it onto a floured surface. Cut off a small golf-ball sized piece for the stems later, then divide the rest into 12 equal pieces and roll each into a smooth ball.
Place your dough balls on a lined baking sheet, cover loosely, and let them rest for 15 minutes. They'll puff slightly, which makes them easier to wrap. Meanwhile, preheat your oven to 350 degrees Fahrenheit and toss your string in vegetable oil.
Pick up a dough ball and wrap a piece of oiled string around its middle, pivoting about 45 degrees each time you wrap, until the string divides the ball into eight sections. Tie it off and set the roll back down, knot-side facing the tray.
Flatten the reserved dough and slice it into 12 small stem shapes with your pizza cutter. Tuck one stem next to each wrapped roll so it's ready to place on top after baking.
Cover the rolls and let them rise another 10 to 15 minutes, until the dough is gently pushing out between the strings. Bake for 22 to 24 minutes, until the tops are a deep golden brown.
Let the rolls cool for at least 20 minutes before snipping the string with kitchen scissors. Once unwrapped, push a stem into the top of each roll, and your pumpkin shaped bread buns are ready to serve.