Grate an 8 oz. block asiago cheese on the side of a box grater with the largest holes. Transfer 1 cup (2 oz.) to a bowl, cover, and place in the fridge. This will be for topping the bagels.
In the bowl of a stand mixer, stir together ½ cup active sourdough starter, 1 cup + 2 tablespoons water, 2 tablespoons honey, and 1 teaspoon fine sea salt with a spatula. Add 4 cups + 2 tablespoons bread flour and the grated asiago cheese, and use your hands to combine the ingredients as well as possible so there are no dry bits left in the bowl.
Place the dough hook on the mixer and mix the dough on number 3 speed for 6-7 minutes. (Or 10 minutes by hand.) Cover the bowl and let rest at room temperature for 8-12 hours.
Line a baking sheet with parchment paper. Turn the dough out onto a clean work surface and divide it into 8 equal pieces. 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 the bagel dough up, gently shape the ring, and place it on the parchment paper. Repeat with remaining dough balls. Cover the dough with a towel and let rise until puffy. This can take 1-2 hours depending on the temperature of your kitchen.(See notes about parchment paper.)
Preheat your oven to 425°F (218°C), ensuring the oven rack is in the center position.
Bring 6 cups of water to a boil in a large stockpot and add one tablespoon of sugar. Place 2-3 bagels into the water, ensuring not to crowd the pot. If they sink to the bottom, give them a gentle nudge with a mesh strainer to help them float to the top. Boil them for 2 minutes on each side.
Use a mesh strainer or slotted spoon to remove the bagels and transfer them to a parchment-lined baking sheet. Sprinkle the top of the bagels with the remaining cheese.
Bake the bagels at 425°F (218°C) for 25-28 minutes or until golden brown. Once the bagels have cooled completely, slice in half and serve.