In a heavy, 4-quart saucepan, heat the oil over low heat. Add the garlic and red pepper and saute for 1 minute, just until soft and fragrant. Take care not to scorch the garlic, as this will produce a bitter flavor.
Add the chicken stock, increase the heat, and bring to a boil. Reduce the heat to low and simmer, uncovered, for 1 1/2 hours, or until the garlic flavor has sweetened and mellowed. If at the end of simmering, too much liquid has evaporated, add enough water to bring the level up to at least 8 cups. Towards the end of the cooking time, whisk in the little pieces of bread for thickening, if using (the egg will also thicken the soup, so the bread step is not necessary, just optional).
Add the green onions and cilantro and continue to simmer until the green onions have wilted, 1 to 2 minutes. Adjust seasonings with salt and pepper, to taste, and remove from the heat. Place the egg yolks in a small bowl and whisking constantly, slowly add 1/4 cup of the hot broth to the egg yolks. Continue whisking the egg mixture while slowly pouring 3/4 cup more of the hot broth into the egg mixture. Whisk the entire egg mixture slowly into the hot soup, until fully incorporated.
Ladle the soup into 6 bowls and serve immediately. Garnish each bowl with 1 tablespoon of the crumbled cheese and a cilantro sprig.