Pat chicken dry and season generously with salt and pepper. Using the oil in a nonstick skillet over medium-high heat cook chicken skin side down until well browned. Flip and continue cooking until another side is browned; remove chicken to a platter and set aside.
Place onion in the now empty skillet, add a pinch of salt and cook until onion starts to brown. Add the garlic and thyme and cook until fragrant, and then mix in the eggplant and squash.
Add the tomatoes and broth to the mixture and work the chicken into the sauce. Cook, stirring occasionally, on simmer until chicken is done. (185F)