Trim the dark green tops and root ends off the leeks. Cut them in half lengthwise, then slice them into 1/2-inch (1.25 cm) thick pieces. Place the sliced leeks in a large bowl of cold water, swishing them around to remove any dirt. Lift the leeks out and dry them well in a salad spinner or on a clean kitchen towel.
Place the chicken breasts and bay leaf in a small saucepan. Add enough water to barely cover the chicken. Bring the water to a boil over high heat.
Reduce the heat to low, cover, and gently simmer for about 10 minutes, or until the chicken reaches an internal temperature of 165°F (74°C). Remove the chicken to a cutting board to cool, then dice it into bite-sized pieces. Strain and reserve the poaching liquid.
Heat the olive oil in a large soup pot or Dutch oven over medium heat. Add the cleaned leeks, chopped garlic, salt, and pepper. Stir well, cover the pot, and cook for 7 to 8 minutes until the leeks soften.
Pour the white wine into the pot and cook for 1 minute to reduce slightly. Add the chicken stock and the reserved chicken poaching liquid. Bring the mixture to a boil.
Reduce the heat to a simmer. Stir in the diced chicken, egg pasta, fresh thyme, and lemon zest. Cook at a gentle boil until the pasta is tender, following the package directions for timing.
Remove the pot from the heat. Stir in the fresh parsley. Taste the soup and add more salt and pepper if needed before ladling into bowls.