In the bowl of a stand mixer, whisk together flour, sugar, salt, and yeast by hand. Add 1¼ cups (300 grams) warm water, oil, 2 tablespoons (42 grams) 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 dough pulls away from bottom and sides of bowl, 8 to 10 minutes. (Dough will be elastic but may not look completely smooth.) Turn out dough onto a lightly floured surface, and shape into a smooth ball.

Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 45 minutes to 1 hour.

Lightly spray 2 baking sheets with cooking spray. Cut 10 (5-inch) squares of parchment paper, and place on prepared pans.

Divide dough into 10 portions (about 115 grams each). Shape each portion into a ball. Place 1 ball in your hand. Using your thumb and forefinger, pinch a hole in center of ball, and gently stretch and roll dough using your fingers to create a 3-inch-wide circle in center of dough. Place on prepared parchment. Repeat with remaining dough. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 15 to 30 minutes.

Preheat oven to 400°F (200°C). Line 2 baking sheets with parchment paper. Place desired toppings in separate wide, shallow dishes.

In a large stockpot, heat 8 cups (1,920 grams) water and remaining ½ cup (170 grams) honey over medium-low heat until just simmering. (Do not boil.) Using the parchment squares as handles, carefully drop bagels, 1 or 2 at a time, into honey water, and discard parchment; cook for 30 seconds per side. Remove from water using a slotted spoon. Carefully place each bagel into desired topping, coating bottom, top, and sides well. Transfer to prepared pans.

Bake until golden brown, 13 to 16 minutes. Let cool on pans for 10 minutes. Serve warm, or let cool on wire racks. Store in airtight container for up to 3 days.