In a small bowl, mix warm water, dry active yeast, and brown sugar. Stir and set aside to allow the mixture to become bubbly and foamy.
In a large bowl or the bowl of a stand mixer, combine 4 1/2 cups of bread flour and salt.
Add the yeast mixture to the flour mixture.
If using a stand mixer, attach the dough hook and knead for 10 minutes, adding up to an additional 3 tablespoons of flour as needed to keep the dough soft yet workable.
If kneading by hand, mix the wet and dry ingredients until a rough dough forms. Turn out the dough onto a clean surface and knead by hand for about 10 minutes, using a push, turn, and fold technique. Add an extra 2-3 tablespoons of bread flour as necessary.
After kneading, let the dough rest for 5 minutes.
Use a sharp knife or bench scraper to divide the dough into 8 equal pieces.
Flatten each portion of dough into a rectangle. Spread a thin layer of butter onto the rectangle, leaving about 1/2 inch at the top of the rectangle plain.
Sprinkle and press approx 2-3 tbsp of brown sugar onto the buttered portion of the dough. Sprinkle cinnamon onto the brown sugar.
Roll up the bagel, starting with the side opposite of the side that has been left empty. Roll tightly and then brush a little water onto the empty dough to help seal the dough.
Take the rolled up dough and bring the ends around to a circle and pinch and slightly knead the ends together. Repeat with remaining dough.
Place on a parchment or silicone-lined baking sheet.
Cover the bagels with plastic wrap and let them rise for 15 minutes.
Preheat the oven to 425°F (220°C) after the bagels have risen.
Boil about 4 inches of water in a large pot. Carefully place the bagels in the water, working in batches. Boil each bagel for 1 minute on each side.
Use a slotted spoon or spider strainer to remove the bagels from the water and place them back on the baking sheet.
While boiling the next batch of bagels, brush the first batch with an egg wash mixture.
After boiling all the bagels and brushing with the egg wash, place them in the preheated 425°F (220°C) oven. Bake for 15-17 minutes, or until they turn golden brown.
Once baked, remove the bagels from the oven and let them cool on the baking sheet for 10 minutes. This allows them to set and makes them easier to cut.