Whisk the starter, pumpkin puree, brown sugar, and water together in the bowl of a stand mixer. Then add the flour, salt, and pumpkin pie spice, mixing until a shaggy dough forms. Knead for 2 to 3 minutes until smooth and stiff.
Cover and let rise until puffy and doubled, about 6 to 8 hours at room temperature. For deeper flavor, you can let the dough rise at room temperature, then move it to the fridge overnight and shape the next day.
Divide into 8 pieces. Roll each into a smooth ball, then poke your thumb through the center and stretch the hole wider than feels natural, about the size of a walnut. It will shrink back during boiling and baking.
Place shaped bagels on a parchment lined tray, cover, and let rest until puffy, about 60 minutes.
Add the honey to a large pot of boiling water. Boil the bagels for 30 to 60 seconds per side. T
Brush each boiled bagel with egg wash and add your topping of choice. Bake at 425F for 20 to 25 minutes, rotating the pan halfway through, until deep golden brown on top and bottom.