If you are using frozen blueberries, thaw them before making the blueberry puree. You will use the blueberries and the liquid from the thawing process. Place the blueberries, and liquid if using thawed blueberries, in a food processor and pulse to break up the flesh, leaving some larger pieces.
In the bowl of a stand mixer, stir together the blueberry puree, starter, water, honey, and salt with a spatula. Add the bread flour and use your hands to bring the ingredients together as best as possible. The dough will be stiff. Cover the bowl and allow the dough to rest for 1 hour.
Place the dough hook on the mixer and mix the dough on the lowest speed for 6-7 minutes. (Or 6-7 minutes by hand.) If the dough is too hydrated from the blueberries, add 25 grams of flour at a time until the dough comes together and is not sticky. Cover the bowl and let rest at room temperature for 8-12 hours.
Line a baking sheet with parchment paper. Turn the dough onto a lightly floured work surface and divide it into 8 equal pieces, approximately 120 grams each. Shape each piece into a ball. To shape the bagel, place a dough ball on the work surface and use your thumb to poke a hole straight down through the middle of the dough. Pick up the bagel dough and gently shape the ring by twirling the inside of the bagel with your two pointer fingers while stretching it. Place it on the parchment paper. Repeat with remaining dough balls. Cover the dough with a towel and let rise for 30-60 minutes or until puffy. (See notes about parchment paper.)
Place the oven rack in the center position and preheat your oven to 425°F (218°C). Bring 6 cups of water to a boil in a large stockpot.
Cut the parchment paper into squares and use the paper to drop the bagel dough into the boiling water. After the bagel has been in the water for 30 seconds, the paper will peel off easily. Boil for 2 minutes per side. Use a slotted spoon to transfer them to a parchment-lined baking sheet.
Bake the bagels at 425°F (218°C) for 25 minutes or until golden brown.