In a large mixing bowl or stand mixer, whisk together the active sourdough starter, pumpkin purée, water, honey, and pumpkin pie spice until smooth. Add the bread flour and salt. Knead on medium speed for 7 to 8 minutes until a smooth ball forms. The dough will be soft and slightly sticky but should pull away from the sides of the bowl. Transfer the dough to a lightly oiled large bowl, cover with plastic wrap, and let it rest at room temperature for the first rise (bulk fermentation), about 6 to 8 hours, or until doubled in size.
Lightly flour your work surface and turn out the risen dough. Use a kitchen scale or bench scraper to divide the dough into equal pieces, about 8 for standard-sized bagels. Take each piece of dough and roll it into a smooth ball. Using your thumb, poke a hole in the center of each dough ball and gently stretch to form a classic bagel shape. Place the shaped bagels on a parchment-lined baking sheet and cover them loosely with a plastic bag or plastic wrap to prevent drying out. Allow them to rest for 30 minutes.
For deeper flavor and easier shaping, place the bagel dough in the refrigerator overnight. Slow fermentation develops a stronger sourdough flavor and makes the bagels easier to handle the next day.
Fill a large pot of water and bring it to a gentle boil. Add 1 tablespoon of baking soda. Using a slotted spoon, carefully lower 1 to 2 bagels at a time into the boiling water. Boil for 30 to 40 seconds per side for softer bagels or up to 1 minute per side for a chewier texture. Return bagels to a parchment-lined baking sheet or baking trays as you finish boiling them.
Preheat your oven to 425°F (220°C). Bake the bagels on the middle rack for 20 to 25 minutes, until beautifully golden brown. Transfer to a cooling rack and let them rest for at least 30 minutes before slicing.
In a small mixing bowl, combine the softened cream cheese, butter, maple syrup, cinnamon, and salt. Whip until smooth and creamy.