Mash the potato. Bring 2 1/2 quarts water to a boil in a large pot. Add the potato and cook until tender, 20 to 25 minutes. Remove the potato, reserving the cooking water. When cool enough to handle, peel the potato and mash in a medium bowl until smooth, adding a few drops of the cooking water, if needed.
Make the dough. Stir the olive oil and honey into the mashed potato. Mix the flour, 2 1/2 teaspoons salt and the yeast in the bowl of a stand mixer fitted with the paddle attachment.
Add the potato mixture and 1 cup reserved cooking water to the mixer; beat on medium speed until stiff but malleable, 10 minutes. Beat in a few more tablespoons cooking water, if necessary. (Reserve the remaining cooking water.)
Knead the dough. Turn the dough out onto a lightly floured surface. Knead by hand until the dough is smooth and bounces back when pressed, about 5 minutes.
Let it rest. Drizzle a little olive oil into a large bowl. Add the dough and turn to coat it with oil. Cover the bowl with a damp dishcloth and set aside in a warm spot until the dough is doubled in size, 1 1/2 to 2 hours.
Cook the onion. While the dough rises, melt the butter in a large skillet over low heat; add the onion and cook until transparent and soft but not browned, 20 to 30 minutes. Season with 1/2 teaspoon salt and set aside.
Form the bagels. Punch down the dough. Form into a log, then cut into 12 equal pieces. Roll each piece into a tight ball and set on a parchment-lined baking sheet; cover with a damp cloth and let rise 30 minutes. Make a hole in the center of each ball with your thumbs. Stretch and turn the ring until the hole is about 2 inches wide. Return to the baking sheet, cover with a damp cloth and let rise 20 more minutes. Preheat the oven to 450 degrees F.
Boil the bagels. Bring the remaining reserved potato cooking water to a boil. Carefully drop the bagels into the water, a few at a time, and cook 30 seconds. Flip the bagels with a slotted spoon and cook 30 more seconds. Set the bagels on a rack to drain, then return to the parchment-lined baking sheet.
Bake the bagels. Transfer the baking sheet to the oven; bake until the bagels start to brown, about 10 minutes. Beat the egg with 1 tablespoon water; brush the mixture on the bagels and top with the sauteed onion. Continue baking until golden, 10 to 15 more minutes. Let cool 15 minutes before serving.