In a medium soup pot, bring the chicken stock and 2 cups water to a low boil.
Meanwhile, in another soup pot, heat the EVOO, 2 turns of the pan, over medium-high heat. Add the chicken and cook, stirring, until lightly browned, about 5 minutes. Add the thyme and bay leaf, then stir in the onion, carrot and mushrooms; season with salt and pepper. Cover the pan and cook the vegetables until softened, 6 to 7 minutes.
Pour the hot stock over the vegetables and bring to a boil. Stir in the orzo and cook until al dente, about 8 minutes. Remove from the heat and add the arugula and basil to wilt. Stir in the lemon peel, discard the bay leaf and season with salt and pepper. Serve with lots of grated cheese at the table.