Heat 2 tablespoons oil in a large Dutch oven over medium-high heat. Season trimmed chicken evenly on all sides with ¼ teaspoon salt and ¼ teaspoon pepper; place in the Dutch oven, smooth-side down. Cook the chicken until nicely browned, about 4 minutes per side. Remove to a plate. Do not wipe the pot clean. Add chopped onion, chopped carrot and chopped celery; cook, stirring occasionally, until softened, about 4 minutes.

Add sliced cabbage; cook, stirring often, until beginning to soften, about 2 minutes. Add minced garlic and 1 tablespoon garlic-and-herb seasoning; cook, stirring constantly, until fragrant, about 1 minute.

Add 5 cups broth and the remaining ¼ teaspoon salt, scraping up any browned bits from the bottom of the pot. Nestle the chicken into the broth mixture; bring to a vigorous simmer over medium-high heat. Reduce heat to maintain a gentle simmer; cover and cook, undisturbed, until the cabbage is tender and the chicken is cooked through (an instant-read thermometer registers 165°, about 20 minutes. Transfer the chicken to a cutting board.

Add rinsed beans to the pot; cook, stirring occasionally, until the beans are heated through, about 5 minutes. Shred the chicken using 2 forks; return to the pot. Remove from heat, and stir in 1½ tablespoons lemon juice. Divide the soup evenly among 6 bowls and top with ¼ cup pesto.