Combine the water, yeast, and 3 tablespoons of the sugar in the bowl of an upright mixer fitted with a dough hook. Stir and let stand until foamy, about 5 minutes. Gradually add 4 cups of the flour and the salt, and mix until the mixture comes together.

Add 1 to 1 1/2 cups additional flour 1/2 cup at a time to make a stiff dough, either stirring with the wooden spoon or working with your hands. Turn out onto a lightly floured surface and knead until smooth and no longer sticky, about 5 minutes, adding just as much flour as needed. (Dough should be heavier and stiffer than regular yeast bread dough.)

Grease a large bowl with 1 teaspoon of the oil. Place the dough in the bowl, turning to coat. Cover and let rise in a warm, draft-free spot until almost doubled, about 1 hour.

Remove from the bowl and punch down the dough. Divide into 12 equal pieces, about 2 to 3 ounces each, measuring about 4 inches across. Form each piece of dough into a ball. Roll each ball into a 4 to 6-inch log. Join the ends and place fingers through the hole and roll the ends together. Repeat with the remaining dough. Place on a lightly greased surface, cover with a clean cloth, and let rest until risen but not doubled in a draft-free spot, 20 to 30 minutes.

Preheat the oven to 400 degrees F.

Grease a baking sheet with oil.

In a large, heavy pot, bring a large pot of water and the remaining tablespoon of sugar to a boil. In batches, add the bagels to the water and boil, turning, for 30 seconds to 1 minute. Dip the bagel tops in desired toppings. Flip bagels onto the prepared sheet pan. Bake for 5 minutes, turn over and cook for another 30 to 35 minutes.

Remove from the oven and let cool on a wire rack.