In the bowl of a stand mixer, whisk together bread flour, granulated sugar, kosher salt, and instant yeast by hand. Add 1¼ cups of warm water, neutral oil, 2 tablespoons of honey, egg, and egg white; using the paddle attachment, beat at low speed until a shaggy dough forms, about 1 minute.
Switch to the dough hook attachment. Beat at low speed until the dough pulls away from the bottom and sides of the bowl, about 8 to 10 minutes. The dough should be elastic and slightly tacky.
Turn out the dough onto a lightly floured surface and shape it into a smooth ball. Lightly oil a large bowl and place the dough inside, turning to coat the top. Cover with a damp cloth and let rise in a warm place until doubled, for 45 minutes to 1 hour.
After the dough rises, lightly spray two baking sheets with cooking spray. Cut 10 squares of parchment paper, and place them on prepared pans.
Divide the dough into 10 portions, each weighing about 115 grams, and shape each portion into a ball. Pinch a hole in the center of each ball and stretch it gently to create a 3-inch-wide circle.
Place the shaped bagels on the prepared parchment and cover them to rise again for 15 to 30 minutes until puffed.
Preheat your oven to 400°F (200°C). Prepare your toppings in shallow bowls.
In a large stockpot, heat 8 cups of water and the remaining honey over medium-low heat until just simmering. Using the parchment as handles, drop the bagels into the honey water, cooking for 30 seconds on each side.
Remove the bagels from the water using a slotted spoon and place them into your desired topping, coating well. Transfer to the pans.
Bake the bagels until golden brown, about 13 to 16 minutes. Let them cool on the pans for 10 minutes before serving warm or cooling completely on wire racks.