Prep and Mix the Dough: Preheat the oven to its lowest setting (usually 170°F / 77°C) for 10 minutes, then turn the oven off. In the bowl of a stand mixer, whisk together the bread flour, brown sugar, instant yeast, pumpkin spice, and salt. Make a well in the center. Add the pumpkin purée, room-temperature butter, and the egg.
Add Warm Liquid and Knead: In a microwave-safe container, mix the heavy cream and water. Heat for 30 seconds until it reaches 110°F (43°C). (Do not exceed 115°F or you will kill the yeast!) Pour the warm liquid into the bowl. Attach the dough hook and mix on low to combine, then increase to medium-low and knead for 10 minutes until the dough is smooth and no longer sticky.
The First Rise: Grease a large bowl with 1 teaspoon of oil. Shape the dough into a tight ball, place it in the bowl, and roll it to coat in the oil. Cover loosely with plastic wrap and a clean hand towel. Place the bowl in the warmed, turned-off oven to proof for 1 hour. (No peeking!)
Portion the Dough: Turn the oven back to 170°F (77°C) for 10 minutes, then turn off. Punch down the risen dough and turn it out onto a lightly floured surface. Shape it into an even log, then cut it into 10 equal pieces. Keep the dough covered with a towel while working.
The Twine Trick: Pinch and roll one piece of dough into a perfectly smooth ball. Lay one 40-inch string of butcher's twine flat on your counter. Place the dough ball in the center. Bring the ends up and tie them loosely across the top. Rotate the string 45 degrees clockwise, flip the dough over, and cross the string again, repeating until you have 8 evenly spaced segments. Tie a loose bow to secure it and trim the ends. (Do not tie tightly; the dough needs room to expand!) Repeat with all 10 rolls.
Second Rise: Place the tied dough balls onto a parchment-lined baking tray, spaced 2 inches apart. Cover with a towel and place back into the warmed oven for a second proof of 30 minutes, or until doubled in size. Remove the tray from the oven.
Glaze and Bake: Preheat the oven to 350°F (177°C). Whisk 1 egg with 1 teaspoon of water. Brush the egg wash over the entire surface of each bun. Bake on the middle rack for 20 to 25 minutes until golden brown. Let cool completely on a wire rack, then carefully snip and remove all the twine. Press a pecan or walnut half into the top center of each bun to create a stem. Serve warm!