Pat chicken thighs dry with paper towels. Season with 0.5 teaspoon salt and the pepper. Heat olive oil in a large skillet over medium-high heat. Add chicken skin-side down and cook without moving until deeply golden, 5 to 6 minutes. Flip and cook 2 minutes more. Transfer to a plate; the chicken will not be cooked through.

Pour off all but 1 tablespoon fat from the skillet. Add onion and bell pepper; cook over medium heat until softened and lightly browned at the edges, 4 to 5 minutes. Stir in garlic and cook until fragrant, 30 seconds. Scrape this mixture into the slow cooker.

To the slow cooker, add eggplant, zucchini, crushed tomatoes, tomato paste, herbes de Provence, and remaining 0.5 teaspoon salt. Stir well to combine. Nestle the seared chicken thighs on top, skin-side up, pouring in any accumulated juices.

Cover and cook on LOW for 6 to 7 hours or HIGH for 3 to 4 hours, until chicken is fall-off-the-bone tender and vegetables are soft but not mushy. The eggplant should collapse into the sauce and the chicken should register 165 degrees F on an instant-read thermometer.

Taste the sauce and adjust salt if needed. If the sauce seems too thin, remove the chicken and simmer the vegetables on HIGH with the lid off for 15 minutes. Serve hot, scattered with fresh basil, alongside crusty bread or over polenta.