Combine dry ingredients. In the bowl of a stand mixer, add the flour, salt, cinnamon, pumpkin pie spice, and brown sugar. Whisk until thoroughly combined.
Add wet ingredients. Add the active potato flake sourdough starter, warm water, and pumpkin puree.
Knead the dough. Using the dough hook attachment, mix on low speed until the ingredients are combined. Increase to medium speed and knead for 5–10 minutes, until the dough pulls cleanly away from the sides of the bowl but still sticks to the bottom.
First fermentation. Remove the dough from the mixer and transfer it to a lightly oiled bowl. Lightly oil the top of the dough to prevent drying. Cover with plastic wrap and ferment at room temperature until doubled in size, approximately 6–10 hours, depending on room temperature and starter strength.
Divide the dough. Once doubled, remove the dough from the bowl and place it onto a clean work surface. Use a bench scraper to divide it into 8 equal portions.
Shape the bagels. Roll each portion into a smooth ball, using the counter to create surface tension. Poke a hole through the center of each ball and gently stretch the opening using your fingers. Spin the dough around your index fingers to widen the hole evenly.
Second rise. Place each bagel on an individual square of parchment paper and arrange them on a baking sheet. Cover loosely with plastic wrap and allow them to rise until soft and puffy, approximately 1–2 hours. Do not overproof.
Prepare the boiling water. Once the bagels are puffy, preheat the oven to 450°F. Bring 12 cups of water to a gentle boil in a large Dutch oven or stockpot. Add the baking soda and brown sugar.
Boil the bagels. Using the individual parchment squares, carefully transfer a few bagels at a time into the boiling water. Allow the parchment paper to release from the dough and remove it. Use the chop stick or something similar to stick into the bagel hole and spin the bagel around a few times. This helps prevent the hole from closing up all of the way. Boil the bagels for 1 minute per side.
Transfer to baking sheet. Using a slotted spoon, remove each boiled bagel and place it onto a parchment-lined baking sheet.
Bake. Once all bagels have been boiled, bake at 450°F for 10–12 minutes, or until deep golden brown and baked through.
Cool and enjoy. Remove from the oven and allow the bagels to cool before slicing. Serve with butter, cream cheese, or your favorite spread.