Add 4 cups of all-purpose flour, 2 tbsp of honey, 2 tsp of dry yeast, 1 tbsp of olive oil, 1½ tsp of salt, 1 egg, and about ¾ cup of lukewarm water into a big bowl or into your stand mixer bowl. If you use a stand mixer, use the dough hook to mix all ingredients so the dough starts to come together before you put the mixer to work. If you use your hands, begin kneading the dough. Note: Add about ¾ of a cup (158 ml) and check if your dough needs any more water. The dough may not be sticky! When the dough passes the windowpane test, is smooth, and dry (not sticky!) you can stop kneading. Take the dough out of the mixing bowl and divide it into 12 portions.
Form small dough balls, lay them on a floured countertop or some baking paper, and cover with a tea towel. Let them rest for about 25 minutes.
After 25 minutes, uncover the balls of dough and make a small hole in each of them with the handle of a wooden spoon. Cover with a tea towel again and let them rest for 35 minutes.
Preheat your oven to 437°F or 225°C, and put a large pot with water on the stove over high heat. When the water is boiling, lower the heat so the water will simmer, not boil.
After the second rest, take a few bagels (I recommend not more than 3 in each run) and put them in the simmering water. Poach the bagels for 1 minute per side.
After the bagels are poached, take them out of the water with a skimmer and put them on a rack to dry up.
Take the second egg, and divide the yolk and egg white. Put the poached bagels on a baking dish with parchment paper, and use a brush to cover the bagel with egg yolk. Sprinkle the bagels with some sesame seeds or other toppings, and bake them for 15 minutes in the oven until they are golden brown.
Take the bagels after 15 minutes out of the oven and let them cool a bit before serving on a rack.