Bring 1 1/2 cups water to a boil and add the broccoli, cooking on medium-high heat for 3 to 4 minutes. Do not overcook or broccoli will lose its bright green color and turn bitter.
Puree the broccoli and cooking water in a blender or food processor (won’t become completely smooth). Set aside to cool slightly.
Whisk together the flours, yeast, salt, and vital wheat gluten in a 5-quart bowl, or in a lidded (non-airtight) food container.
Add the 1 3/4 cups water and broccoli puree and mix without kneading, using a spoon, Danish dough whisk, 14-cup food processor, or heavy-duty stand mixer (with paddle). You might need to use wet hands to get the last bit of flour to incorporate if you’re not using a machine.
Cover (not airtight) and allow to rest at room temp until it rises and collapses (or flattens on top), about 2 hours. Dough can then be used immediately, although it’s easier to handle when cold. Refrigerate in a non-airtight lidded container and use over the next 7 days.
On baking day, grease a muffin pan. Dust surface of refrigerated dough with flour and cut off a 1 1/2 pound piece (cantaloupe-sized) piece. Dust the piece with more flour and quickly shape a loose ball. Form the buns by dividing the ball into 8 roughly equal portions, each about the size of a plum. Cut each of these into 4 smaller pieces. Dust and shape each one into a ball and put the 4 balls together and fit them into the cups of the muffin tin. Repeat process with remaining dough two times. Each 1 1/2 pounds of dough makes 8 buns.
Allow to rest, loosely covered with plastic wrap, for 40 minutes (or 20 minutes if you’re using fresh unrefrigerated dough). Refrigerated dough doesn’t rise much in this resting phase.
If you’re using a baking stone, preheat the oven to 450 degrees F, 30 minutes before baking time. If you’re not using one, a 5 to 10 minute preheat is enough. Place an empty broiler tray on any other rack that won’t interfere with the rising buns.
Just before baking, sprinkle the buns with the grated cheese, being careful not to get it on the pan. Depending on the shape and depth of your muffin cups, you may need less than we call for in the Ingredients list. I used less than a tablespoon of cheese per bun.
Place the muffin tin directly on the hot stone (or a middle shelf of the oven). Pour 1 cup of hot tap water into the broiler tray, and quickly close the oven door. Bake for about 20 to 25 minutes until the cheese is nicely browned. Remove buns from the pan and allow to cool slightly before eating.