Weigh the flour, water, sourdough starter, salt, and 10 g granulated sugar into a mixing bowl. Mix roughly until no dry flour remains.

Add the olive oil to the dough. Mix until incorporated.

Knead the dough for about 2 minutes until smooth. Do not knead until a windowpane develops.

Lightly oil the mixing bowl and place the dough inside. Cover the bowl with a clean kitchen towel.

Let the covered dough bulk-ferment at room temperature until doubled in size. Allow extra time when using sourdough discard.

Deflate the risen dough and divide it into 6 equal pieces. Shape each piece into a smooth ball.

Rest the dough balls for 5 minutes. Cut 6 parchment paper squares while they rest.

Roll one dough ball into an 8 x 4 inch rectangle. Spread a small amount of Gruyere cheese over the dough without overfilling.

Roll the dough rectangle lengthwise into a tube and pinch the seam closed. Leave one end open.

Stretch the open end slightly and tuck the other end into it. Wrap into a ring and pinch the ends firmly together.

Set the shaped bagel on one parchment square. Repeat the shaping process with the remaining dough balls and cheese, then place all bagels on the baking sheet.

Cover the shaped bagels and let them rise for about 2 hours until visibly puffy.

Preheat the oven to 420°F (215°C). Fill the large pot with water and bring it to a boil on the stovetop.

Stir the remaining 1/4 tablespoon granulated sugar into the boiling water. Keep the water at a gentle boil.

Boil the risen bagels for 10 to 20 seconds per side. Lift them from the water with a slotted spoon and return them to the baking sheet.

Bake the bagels at 420°F (215°C) for 20 minutes. Remove them when deeply golden brown.

Cool the bagels on the baking sheet for at least 10 minutes before serving. Serve warm or at room temperature.