Heat oil in a 12-inch skillet over medium heat until shimmering. Add onion, thyme, tomato paste, garlic, anchovy, and red pepper flakes and cook until onion is softened and lightly browned. Stir in flour and cook for 1 minute. Slowly whisk in vinegar, scraping up any browned bits and smoothing out any lumps; cook until slightly reduced, about 3 minutes, and transfer to a slow cooker.

Stir tomatoes, broth, wine, and bay leaves into slow cooker. Season chicken with salt and pepper and nestle into slow cooker. Cover and cook until chicken is tender, 4-6 hours on low.

Gently stir in chard, cover, and cook on high until tender, 20-30 minutes. Transfer chicken to serving platter and tent loosely with aluminum foil. Let braising liquid settled for 5 minutes, then remove fat from surface using a large spoon. Discard bay leaves. Season with salt and pepper to taste. Spoon 1 cup sauce over chicken and serve with remaining sauce.