Place the chicken in a small pot and add enough water to just barely cover, 2 1/2 to 3 cups. Add the quartered onion and the bay leaf, cover and bring to a boil. Uncover the pot, lower the heat and simmer to cook through, about 10 minutes. Transfer the chicken to a bowl, then shred with 2 forks. Reserve the cooking liquid.
Meanwhile, in a large saucepan, melt the butter over medium heat. Add the orzo and cook, stirring often, until golden, 2 to 3 minutes. Stir in the rice, then add 2 cups chicken stock and bring to a boil. Lower the heat to low, cover and cook until the rice is tender, about 15 minutes. (Cooking the rice separately from the soup lets you add it to the liquid when youre ready to eat, so it doesnt overcook.)
In a medium soup pot, heat the EVOO, 1 turn of the pot, over medium-high heat. Add the chopped onion, celery and carrot; season with salt and pepper. Cook until the vegetables are tender, about 5 minutes. Strain in the reserved cooking liquid and the remaining 2 cups chicken stock and bring to a boil. Stir in the thyme, dill, parsley and lemon peel.
To serve, pile the chicken and rice in bowls and top with the broth.