Whisk together the lemon juice and eggs in a medium liquid measuring cup or mixing bowl; set aside.
Heat butter and oil in a large Dutch oven set over medium-high heat. Add onions, carrots and celery. Sauté for 4-5 minutes, stirring occasionally, until softened. Add garlic and sauté for 30 seconds, until fragrant.
Stir in chicken broth, lemon zest, and herbs. Bring the soup back to a gentle simmer, then reduce heat to medium-low to maintain.
Add the rice, cover, and continue to simmer for 18-20 minutes, until the rice is almost tender.
Remove the pan from the heat and temper the egg mixture. Whisking constantly, slowly drizzle about ¼ cup of the hot soup into the egg mixture. Repeat 3 times, continuing to whisk, until the mixture is smooth.
Whisk the tempered egg mixture into the soup, whisking constantly. Remove and discard the bay leaf.
Stir the shredded turkey and fresh parsley into the soup. Season to taste with additional salt and pepper, if desired. Garnish with additional fresh herbs before serving.