Place 1 tablespoon of chicken stock in a saucepan over medium-low heat, and cook and stir the onion and garlic in the broth until they are softened, about 3 minutes. Pour in the rest of the broth and 1/2 cup of milk, and whisk in the salt, black pepper, parsley, lemon pepper seasoning, allspice, and paprika. Bring the mixture to a boil, reduce heat, and allow to simmer for 1 to 2 minutes.
Whisk together the remaining milk and the flour until smooth, and whisk the milk mixture into the hot stock mixture. Keep whisking continuously to avoid lumps until the soup comes almost to a boil and thickens.