Place water, flour, sugar, and salt in the bread machine (in that order). Make a small well in the top of the flour and pour yeast inside, then cover well with the flour.
Set bread machine on the "dough" setting, which will mix the dough and allow it to rise.
When setting is done, punch down the dough in the pan and let it rest for 10 minutes.
Remove dough from bread machine and divide it into eight equal pieces, then shape each into a dough ball by rolling it gently in a circular motion against the (un-floured) counter.
Coat your finger in flour and press into the center of each dough ball to form a ring, gently stretching the ring into a bagel shape. Cover bagel shapes with a damp kitchen towel and allow to rest for 10 minutes.
Preheat the oven to 425 degrees and bring a large pot of water to a boil.
Place bagels 2-3 at a time into the boiling water with a slotted spoon (or by gently lowering them into the water).
Let bagels sit in boiling water for 1-2 minutes, then flip and allow to sit in water for an additional 1-2 minutes (the longer they are in the water, the chewier the texture will be).
Remove from water and place on baking sheet lined with a silicone mat. Brush bagels with egg wash (1 beaten egg plus 1 tablespoon water). Add toppings if desired.
Bake for 20 minutes or until golden. Remove from baking sheet to a cooling rack.