Place all the starter ingredients in a medium-size bowl, and whisk until well combined. The mixture will be thick and shapeless. Cover and set the bowl aside in a warm, draft-free location to rise until doubled (about 40 minutes).
Make the dough.
Once the starter has finished rising, place the bread flour and yeast in the bowl of your stand mixer, and use a handheld whisk to combine well. add the salt, and whisk to combine.
Add the risen starter, butter, and Parmigiano-Reggiano cheese to the bowl, and mix on low speed with the dough hook until combined. Raise the mixer speed to medium and knead for about 5 minutes. It will be sticky dough, but will become stretchier and appear thinner as the mixer kneads it. Spray a silicone spatula lightly with cooking oil spray, and scrape down the sides of the bowl.
Transfer the dough to a lightly oiled bowl or proofing bucket large enough for the dough to rise to double its size, and cover with an oiled piece of plastic wrap (or the oiled top to your proofing bucket).
Place the dough in a warm, draft free location to rise until nearly doubled in size (about 1 1/2 hours).
To make the dough easier to handle, after it has finished rising, place the sealed dough in the refrigerator to chill for 30 minutes before handling.
Shape the bagels.
Once the dough has finished its first rise, line a large, rimmed baking sheet with unbleached parchment paper, grease it lightly with cooking oil and set it aside. Turn the dough onto a lightly floured surface. Knead until smooth by scraping the dough over on itself using a cake turner or bench scraper, sprinkling with more gluten free bread flour lightly as necessary to prevent sticking.
Divide the dough into six parts. Shape each piece of dough into a round, and then poke a hole with a well-floured finger vertically into the center of the round dough. Move your finger in a circular motion to widen the hole so that it does not close during proofing.
Place each piece of bagel-shaped dough on the prepared baking sheet about 2 inches apart, cover with oiled plastic wrap, and set in a warm, draft-free location to rise until nearly doubled in size (about 45 minutes to 1 hour, depending upon the environment).
About 25 minutes before the dough has finished rising, preheat your oven to 375°F.
Boil the bagels.
Dissolve 1 tablespoon of molasses and 1 teaspoon of kosher salt in 6 cups of water (approximate measurements are fine here). Bring the mixture to a boil over medium-high to high heat.
Once the dough has finished rising, place the shaped bagels a few at a time into the boiling bath for only 30 seconds per side.
Remove the bagels with a strainer and return them to the baking sheet.
Bake.
Brush the boiled bagels with the egg wash, and sprinkle generously with about 3/4 of the grated Asiago cheese. Repeat with the remaining bagel dough.
Place the baking sheet in the center of the preheated oven and bake for 15 minutes. Remove the bagels from the oven and quickly sprinkle the remaining Asiago cheese over the nearly-baked bagels. Rotate the baking sheet, return the bagels to the oven and continue to bake until they are golden brown all over and the cheese is melted (about 5 minutes more). To brown the cheese a bit, place it under your oven’s broiler for about 1 minute.