In a large pot, combine the onion, garlic, chicken, oats, carrot, olive oil, salt, pepper, and water. Partially cover with the lid, and bring to a simmer over medium-high heat.
Lower the heat to medium and simmer for 20 to 25 minutes, until the chicken is cooked through.
Remove the chicken, shred it with two forks, and return it to the pot. Stir well.
Pour in the warm milk slowly, stirring constantly. Simmer over medium-low heat for 10 minutes.
Remove from the heat. Stir in half the parsley and the lemon juice. Taste, and add more salt, pepper, or lemon if needed.
Serve garnished with the remaining parsley, with extra lemon on the side.