In the bowl of a stand mixer add the water and yeast. If using active (not instant) yeast, let sit for 10-15 minutes until bubbly. If using instant yeast, continue right away.
Add flour, salt, sugar and mix. Knead vigorously for 10 minutes (15 minutes if you're doing this by hand). This is a very strong dough and won't settle when the mixer is turned off. Will 'thwap' around and clean the sides and might rock your mixer a bit on anything higher than about 3.
Place the dough in a lightly greased bowl, cover, and let sit to rise in a warm place for 60 to 90 minutes. It might not double but it should get very puffy.
Line two baking sheets with parchment and spray with oil. Yes, parchment and oil. You can thank me later.
Deflate the dough, transfer to a work surface, and divide into 8-12 pieces. I usually do 12, six per sheet. Not as big as store-bought bagels but still a good size.
Working with one piece at a time, work into a smooth, round ball. Place the balls evenly on the parchment. When all are formed, cover with oiled plastic wrap and let sit for 30-40 minutes to puff up a bit.
Meanwhile, preheat oven to 425°F.
Get a wide pan ready on the stovetop with about 3" of water. Add sugars and bring to a gentle boil.
Working with one ball at a time, push your thumb through the middle and then widen the hole to about 2". Put back onto the sheet pan and let rest while working on the rest of them.
When all are formed, put them in the order you made them into the water bath -- probably 3 or 4 at a time, depending on the size of your bagels and the size of your pan -- for about 2 minutes then flip and boil 1 minute longer. You may need to turn the heat back up to keep it at a boil. Remove with a slotted-spoon or a spider and return to the parchment-lined baking sheet with the smoothest side up. Continue until all bagels are boiled.
(If adding toppings, whisk the egg white and water until frothy. Brush on the bagels and add seeds or 'everything' topping.)
Bake for 20-25 minutes. If making plain bagels, take out at about 15 minutes and flip over. May want to rotate the sheets and change racks, too.
Remove from oven and put bagels on a wire rack to cool completely.