Place a large Dutch oven over medium heat and add the olive oil.
Add the diced onion, sliced carrots, and chopped celery to the pot. Sauté for 6–7 minutes until the vegetables are tender.
Stir in the minced garlic and sauté for 1 additional minute until fragrant.
Pour the chicken broth into the pot and bring the liquid to a rolling boil over high heat.
Stir in the dry orzo pasta. Reduce the heat to a medium simmer and cook uncovered for 8–9 minutes, stirring occasionally to prevent sticking, until the orzo is tender.
Reduce the heat to low. Stir in the shredded chicken, fresh lemon juice, lemon rounds, and fresh baby spinach.
Simmer gently for 2 minutes until the chicken is heated completely through and the spinach leaves have wilted.
Remove the pot from the heat. Stir in the chopped fresh dill (or parsley). Season carefully with salt and black pepper to taste.
Ladle the hot soup into bowls, ensuring each serving gets a bright lemon slice on top. Serve immediately.