Dissolve yeast in warm water and honey in a stand mixer or a large bowl; let stand for 5 minutes. Stir in half the flour, salt, and whole egg until well blended. Let rest for 20 minutes.
In a stand mixer with a paddle or flat beater, mix-in the remaining flour, up to 5 cups, allowing the mixer to run for 4-5 minutes. Turn dough out onto a floured surface and knead for one or two turns to form a ball.
Oil a bowl with olive oil, place dough in bowl and turn over so that oiled surface is face-up.
Cover the bowl with plastic wrap and then a towel, and let it rise in a warm spot (above 70°) for 1 hour, or until nearly doubled in bulk. (The oven, with the oven light ON is a great place to let the dough rise if the room is cool.)
Line two baking sheets with parchment paper and create 6 small circles of Parmesan cheese on each sheet for the bagels to rest on--this will create a cheese crust on the bottom side of the bagel. Make sure the circles are far enough apart so the bagels do not touch when rising.
With oiled hands, punch down dough and divide it into 12 pieces on a lightly floured surface (a kitchen scale can be helpful to make sure they are even).
Shape each piece into a round, smooth ball, then punch thumb through middle of dough and stretch to create a 1 1/2" hole in the middle. Place bagel on top of prepared circle of Parmesan and repeat with remaining dough.
Once all the bagels are formed, brush egg white on all exposed sides and sprinkle with remaining cheese. Don't worry about the cheese falling in the hole or any cheese that falls on the pan. Just gently push it up next to each bagel (the bagel will "grab" it as it rises and bakes).
Allow bagels to rise for 20-30 minutes in a warm place away from drafts, or cover with a lightweight smooth towel or tented aluminum foil.
Preheat the oven to 425° and position an oven rack in the lower third of the oven (approximately the third position from the bottom in a seven rack oven) and one in the lowest position.
Pour 2 cups of boiling water in a shallow pan and place on the lowest rack of the oven. Steam will help the crust have the desired "pull" and texture of bagels traditionally boiled in water before baking.
Bake the bagels on the rack above the steam pan for 5 minutes uncovered, then tent with aluminum foil and continue baking 12-15 minutes until light-golden in color.
This will keep the bagels from becoming too browned to toast at a later time. If serving all the bagels right away without toasting, the foil is not needed.
Remove from oven and let cool on the pan for 10-15 minutes. Remove from pan and cool completely on wire racks until ready to slice and serve or freeze.
The amount of flour needed will vary depending on humidity and altitude. The dough is softer and stickier than dough made with white flour.