In a bowl, whisk together flour, salt and pumpkin pie spice.
In the bowl of a stand mixer, whisk together the water, yeast and 2 1/2 tablespoons sugar. Allow it to proof for 5 minutes.
Using a dough hook attachment, mix the sweet potato puree in with the yeast mixture on low speed. Add the flour mixture. Mix on low speed for 5 minutes and then turn the speed up to medium and allow it to mix for an additional minute.
Form the dough into a ball and place in a large, lightly greased bowl. Cover and allow the dough to double in size, approximately 1 hour. Remove the dough and punch it down.
Lightly grease a baking sheet. Using a kitchen scale, measure out 2 ounce balls (these will make small bagels compared to what you would buy in a store. For a more traditionally sized bagel, aim for 3.5 ounces). Roll each ball into a 5-inch log. Join the ends and firmly pinch them together. Once all of the bagels are rolled out, let them rest for an additional 20 minutes.
Preheat the oven to 400°F and bring a large pot of water to a boil with remaining 1/2 tablespoon of sugar. Grease a baking sheet and set aside.
In batches, boil the bagels for around 30 seconds per side.
Remove the bagels with a slotted spoon and place them on the prepared sheet tray. Bake for 12 minutes, flip, and bake for an additional 12 minutes. Allow to cool before serving.