Warm the butter and olive oil in a large soup pot over medium-high heat, then cook the celery, carrots, and onion for 5 to 7 minutes. The vegetables should soften while still holding a little bite, with the onion turning translucent around the edges.
Add the minced garlic and stir for about 30 seconds, then sprinkle in the flour and cook it for about 1 minute. The garlic should smell fragrant quickly, and the flour should coat the vegetables without leaving dry patches.
Slowly pour in the chicken broth while stirring, then add the Italian seasoning and chicken breasts before bringing the liquid to a boil. Keep stirring until the flour disappears into the broth and the liquid looks smooth rather than lumpy.
Set the lid slightly ajar, lower the heat, and simmer the soup for 15 minutes. Small, steady bubbles should rise around the chicken instead of a hard rolling boil.
Add the orzo and cook for about 10 minutes, stirring the pot often with the lid off. The orzo is ready when it looks translucent with a tiny pale center and feels tender with a little firmness.
Transfer the chicken to a cutting board, cut it into bite-size pieces, and return it to the soup with the lemon juice, parsley, salt, and pepper. Turn off the heat before adding the lemon, then taste for a bright, savory broth with a fresh citrus finish.