Step 1: Heat 1 tablespoon of olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Season chicken pieces with 1/2 teaspoon of oregano, salt, and pepper. Sear chicken until golden brown on all sides. Remove chicken and set aside.
Step 2: Add the remaining 1 tablespoon of olive oil to the pot. Add the diced onion and sauté for 5-7 minutes until softened and translucent. Add the minced garlic and cook for another minute until fragrant, being careful not to burn it.
Step 3: Deglaze the pot with white wine (if using), scraping up any browned bits from the bottom. Return the seared chicken to the pot. Pour in the chicken broth, add the bay leaf and remaining 1/2 teaspoon of dried oregano. Bring to a boil, then reduce heat to low, cover, and simmer for 20-25 minutes, or until chicken is cooked through.
Step 4: Remove the chicken from the pot and let it cool slightly. Discard the bay leaf. Once cool enough to handle, shred the chicken using two forks or your hands. Bring the broth back to a gentle simmer. Add the orzo pasta and cook according to package directions (typically 8-10 minutes) until al dente.
Step 5: Return the shredded chicken to the pot. Remove the pot from the heat. Stir in the fresh lemon juice and lemon zest. Taste and adjust seasoning with salt and pepper as needed.
Step 6: Stir in the fresh chopped dill and parsley. Ladle the hot Mediterranean lemon chicken soup into bowls and serve immediately, garnished with extra fresh herbs or a lemon slice if desired.