Prior to making the bagels feed your starter with equal parts cold water and flour. It should be bubbly when you begin making the bagels.
Using a fork whisk the sourdough, yeast, salt, water, and maple syrup in the bowl of your stand mixer.
Once combined transition to the dough hook. Add the flour one cup at a time while mixing on medium speed. Continue adding flour until the dough holds together and is no longer sticky. If you do not know how to recognize this shoot for about 5 cups of flour.
Either using the dough hook or by hand continue kneading until the dough is pliable, around 5 minutes. The dough should be shaped like a ball.
Cover the bowl and allow the dough to rise for around 30 minutes.
Split the dough into the number of pieces of bagels you want. This can make anywhere from 8 larger bagels to 12 smaller bagels well. Personally I make 12 smaller bagels. Allow the balls to rest another 20 minutes.
Prepare a baking sheet or two with parchemnt paper and flour.
Stick your thumb through the middle of each ball of dough rolling the dough up your thumb to form a hole in the middle. If you want a larger hole use your fingers to stretch the hole.
Place each bagel on the baking sheet(s) as you go. Once all the bagels are on the sheets, cover. Allow your bagels to rise for 3-5 hours at room temperature. They should be very puffy. Test a bagel to see if it is properly "proofed" by placing in cold water. If it floats, it is done proofing. If not, allow it to continue rising.
At this point you have two options: you can immediately move to the next step if you want your bagels sooner. Alternatively, for a deeper flavor, cover the dough and leave it in the refrigerator for 12-24 hours. When you are ready to proceed, remove it from the refrigerator and allow the bagels to rise for an additional 20-30 minutes.
Preheat the oven to 400°.
Bring a large pot of water to a rolling boil. Add the baking soda and brown sugar and allow the water to return to a boil.
Gently bagels in the water. Boil on each side for 2-3 minutes. Flip. Once cooked on both sides, remove with a slotted spoon or spatula, shake off any extra water, and place on the baking sheet. At this point add any desired toppings such as Everything but the Bagel. Continue with the remainng bagels.
Bake the bagels for 15-20 minutes. Switch your oven to broil for the final minute.
Allow the bagels to cool before serving.