To make the sponge, in the bowl of a standing mixer, combine the yeast, flour and water. Stir with a wooden spoon until evenly combined. Let rest on the counter at room temperature for 2 hours, until doubled in size. (The dough will be full of large bubbles and very sticky.)
To the sponge, add the additional yeast and place the bowl in a standing mixer fitted with the dough hook attachment. Mix on low until the yeast is incorporated. Add 4 cups of the flour, pumpkin puree, vanilla, granulated sugar, dark brown sugar, pumpkin pie spice, cinnamon, and salt. Begin mixing on low until the flour is incorporated. Increase the mixer to medium speed just until the ingredients begin to form a consistent ball of dough.
Turn the dough out onto a lightly floured surface. Knead the dough for 10 minutes, adding up to an additional 1/2 cup flour as needed to keep the dough from becoming sticky. (I did not need to add any flour to the pumpkin spice dough. The dough should be smooth, firm, and cool to the touch.)
Immediately divide the dough into 16 even pieces (I roll it into a ball and cut it in half, and cut those halves in half, and so on until I have 16 even pieces.) Cover the rolls with a damp towel and allow them to rest on the counter for approximately 20 minutes.
Shape the bagels and refrigerate. Line 2 baking sheets with parchment paper and spray lightly with oil. Holding one piece of dough in both hands, gently press your thumbs through the center to form a hole. Pull gently, working your way around the bagel with your fingers, until the hole is about 2 inches wide (it will close up slightly as the dough rises). Place the bagels 2 inches apart on the baking sheets. Mist them lightly with oil and cover gently with foil or plastic wrap. Let the pans rest at room temperature for 20 minutes.
Refrigerate the bagels overnight. (Note: I have made these without refrigerating them at all and have still found the results to be great.)
Preheat the oven to 500°F with the two racks set in the middle of the oven. Bring a large pot of water to a boil (the wider the pot the better – I use my wok), and add the baking soda.
Remove the bagels from the refrigerator and gently drop them into the water, boiling only as many as comfortably fit without touching. Boil for 2 minutes. Then turn the bagels and boil them on the other side for 2 minutes more. After boiling, return the bagels to the parchment-paper lined baking sheets.
When all the bagels have been boiled, place the pans on two middle shelves in the oven. Bake for approximately five minutes, then rotate the pans to allow them to bake evenly. After the rotation, lower the oven setting to 450°F and continue baking for about 5 minutes, or until the bagels turn light golden brown. (You can continue to bake them for a few minutes more to get a darker, chewier crust.) Allow to cool on a baking rack for 15 minutes before serving.
For storing: I find that these do very well frozen. I allow them to cool completely, slice them, and then store them in tightly sealed plastic bags in the freezer. With a quick toast (5 or 6 minutes) in the oven right out of the freezer, they are fantastic.