Heat the oil over medium-high heat in a large (at least 8-quart), heavy-duty pot or Dutch oven. Add the onion, carrot, celery, and 1/2 Tbs. salt. Cook, stirring occasionally, until the vegetables are softened and lightly browned in places, 4 to 6 minutes. Add the bay leaf or leaves and cook, stirring, about 30 seconds.
Add the broth and bring to a boil. Reduce to a simmer and cook, stirring, until the vegetables are completely tender, 20 to 30 minutes.
Stir in the shredded chicken, the rice, and spinach. Simmer, stirring occasionally, until all are tender and the flavors meld, 5 to 10 minutes.
Stir in the dill and 1 Tbs. of the lemon juice. Taste, and add more lemon juice, 1 Tbs. at a time, if needed, up to 4 Tbs. Season with salt and pepper to taste and garnish each bowl with 1 Tbs. sliced scallion, if you like.