In a blender, combine the tomatoes, onion, chipotle chilies and adobo sauce, garlic and cilantro stems. Blend until smooth, scraping the blender jar as needed, about 2 minutes; set aside.

In a large Dutch oven over medium-high, heat the oil until barely smoking. Add the chicken skin side down and cook, without disturbing, until well browned, about 10 minutes. Transfer to a large plate.

Add the tomato-onion puree to the pot and cook, scraping up any browned bits, until a spatula drawn through the mixture leaves a trail, 6 to 8 minutes.

Stir in the broth, return the chicken to the pot and pour in any accumulated juices. Bring to a simmer, then cover, reduce to medium and cook, stirring occasionally and adjusting the heat as needed to maintain a simmer, until a skewer inserted into the largest thigh meets no resistance, about 30 minutes.

Remove the pot from the heat. Using tongs, transfer the chicken to a clean plate; set aside.

Measure 2¼ cups of the hot broth into a large saucepan and add the rice. Bring to a simmer over medium-high, then cover, reduce to low and cook without stirring until the rice has absorbed the liquid, about 18 minutes. Remove from the heat, uncover, drape a kitchen towel across the pan and re-cover. Let stand until ready to serve.

While the rice cooks, use 2 forks or your hands to shred the chicken into bite-size pieces; discard the skin and bones. Set the shredded meat aside.

Bring the broth in the Dutch oven to a simmer over medium. Add the zucchini and green beans; cook, stirring occasionally, until the beans are tender-crisp, about 5 minutes. Remove the pot from the heat, then stir in the chicken and about half of the cilantro leaves. Taste and season with salt and pepper.