In a small liquid measuring cup, whisk together the whole egg, egg yolk, 1 tablespoon of the olive oil, and 1 tablespoon of water until thoroughly combined. In the bowl of a food processor, add the flour and 1/2 teaspoon salt. Pulse 2 to 3 times to mix. With the motor running, slowly stream in the egg mixture and process until a crumbly dough forms, resembling coarse couscous. It should not feel sticky when pinched. If it’s too dry, add water a tablespoon at a time; if it’s too moist, sprinkle in a bit more flour.

Transfer the dough onto a lightly floured surface and knead briefly, about 4 to 5 turns, until it comes together into a smooth ball. Place the dough in a bowl, cover, and refrigerate for about 30 minutes to chill and firm up (see Cook’s Note if applicable).

Meanwhile, warm the remaining 1 tablespoon of olive oil in a large soup pot over medium-high heat. Add the celery and onion, cooking for 4 to 5 minutes until the vegetables become tender and aromatic. Stir in the grated carrots and continue to cook for another 3 to 4 minutes until softened. Mix in the thyme leaves, grated garlic, 2 teaspoons of kosher salt, and a few turns of black pepper. Sauté for another minute, just until the garlic becomes fragrant.

Pour in the chicken stock and add the bay leaves. Raise the heat and bring the mixture to a boil, which should take about 7 to 8 minutes with occasional stirring.

Lower the heat to a gentle simmer. Using the large holes of a box grater, grate about one-third of the chilled dough directly into the simmering soup. Tap the grater against the pot to help release the freshly grated noodles. Stir frequently to prevent clumping and wipe the grater clean between batches if needed. Repeat the process until all the dough is used.

Let the noodles simmer for about 5 minutes, or until they float and are tender. Add the shredded chicken to the pot and cook for an additional 2 to 3 minutes, just until heated through.

Ladle the soup into bowls and garnish with torn parsley leaves if desired. Serve hot.