In the bowl of a stand mixer combine the warm water and yeast with a dough hook attachment. Let the mixture sit for about 5 to 10 minutes until it gets foamy.
Add in the bread flour and all-purpose flour, sugar, and salt. Mix on low speed for about 2 minutes. The dough will be a bit sticky at this point.
Continue kneading the dough in the stand mixer for up to 20 minutes, or until the dough becomes smooth, shiny, and elastic. If the dough is too sticky add a little more flour.
Transfer the dough to a large greased bowl and cover with plastic wrap or a lid to rise. Allow the dough to rise at room temperature until it doubles in size.
Divide the dough into 4 equal pieces, and shape them into round bagel-shaped discs. Use your thumb or finger to poke a hole into the center of each bagel.