PROOF THE YEAST
In a large mixing bowl, add the warm whole milk, active dry yeast, and 1 tablespoon of the light brown sugar, then whisk by hand until dissolved. Let sit for 5 to 10 minutes, until foamy on top.
MAKE THE DOUGH
To the same bowl, add the remaining light brown sugar, egg, egg yolk, pumpkin puree, and melted butter, then whisk by hand until combined.
Add the salt, pumpkin pie spice, and bread flour to the bowl.
Using a stand mixer fitted with the hook attachment on low speed, or a wooden spoon followed by hand kneading, combine the mixture into a dough, then knead for 6 to 8 minutes until smooth and elastic. The dough should be slightly tacky to the bottom of the bowl; if it feels dry, add a splash of milk as needed.
Transfer the dough to an oiled bowl, cover, and let rise in a warm place for 1 to 1 and 1/2 hours, until doubled in size.
MAKE THE FILLING
In a small bowl, add the room-temperature butter and pumpkin puree, then stir with a spoon until combined.
In a separate small bowl, add the light brown sugar and cinnamon, then stir together until combined.
MAKE THE HONEY GLAZE
In a medium bowl, add the powdered sugar, honey, whole milk, melted butter, and vanilla extract, then whisk by hand until smooth and creamy.
SHAPE AND FRY
Once the dough has risen, flour a work surface and roll the dough out with a rolling pin into a large rectangle, until it is about a little less than 1/2 an inch thick and 14 by 20 inches in width and length.
Spread the butter and pumpkin puree mixture evenly over the dough with a rubber spatula, then sprinkle the brown sugar and cinnamon mixture evenly over that, pressing down firmly.
Starting from one of the shorter ends, roll the dough up tightly into a log, then pinch the seam and ends together tightly to seal.
Using unflavored dental floss, slide it under the log and cross the ends over each other to cut through cleanly without squishing the dough — this is the cleanest method, but a sharp knife works too. Cut the log into 12 slices, about 1- to 1.5- inches thick.
Place each roll on its own parchment paper square, then top with another parchment paper square and lightly roll it out with a rolling pin to flatten slightly, until it's about the size of a donut. Insert a toothpick through the seam of each roll and into the center to keep the end from unrolling — you can leave the toothpick in while frying. Then transfer them to a large tray.
Cover and let proof again for about 30 minutes, until visibly puffy.
Heat vegetable oil in a deep pot or pan to 325°F to 350°F.
Drop the rolls in the hot oil, lifting off the parchment square. Fry them in batches of 3 to 4, making sure not to overcrowd the pan, for about 2 to 3 minutes per side, until deep golden brown.
Transfer to a wire rack to drain any excess oil.
GLAZE AND SERVE
While the rolls are still warm, dunk each one into the honey glaze, letting any excess drip off, then return to the wire rack to set for about 5 minutes, then enjoy!