In a small bowl, mix warm water, dry active yeast, and 1 tbsp 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, 1/4 cup of brown sugar and salt.
Add the yeast mixture to the flour mixture along with the vanilla and butter extract.
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. Add the sprinkles into the dough when the dough is almost combined and then finish the kneading.
If kneading by hand, mix the wet and dry ingredients until a rough dough forms.Add the sprinkles and then 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.
Shape each piece into a tight ball. Then, press your thumb into the center of each ball, gently stretching and rotating the dough around your thumb to form a bagel shape. Place on a parchment or silicone-lined baking sheet.
Cover the bagels with plastic wrap and let them rise for 20 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.
After boiling all the bagels 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.
For storage, place the cooled bagels in a zip-top bag and keep them at room temperature. They will stay fresh for up to 3 days.