Prep the slow cooker base. Add crushed tomatoes, chicken broth, tomato paste, balsamic vinegar, sugar, Italian seasoning, red pepper flakes, salt, and black pepper to the slow cooker. Stir to combine.
Soften the aromatics. Heat olive oil in a skillet over medium.
Sauté onion for 3–4 minutes until translucent. Add garlic and cook 30 seconds until fragrant. Transfer to the slow cooker. (You can skip this step, but it adds better flavor.)
Add the chicken. Nestle the chicken into the sauce, turning to coat.
Make sure the pieces are mostly submerged for even cooking.
Cook low and slow. Cover and cook on Low for 5–6 hours or on High for 2 1/2–3 1/2 hours, until the chicken is very tender and easily shreds with a fork.
Shred and finish. Remove the chicken to a board and shred with two forks. Return it to the slow cooker and stir. Taste and adjust salt, pepper, and acidity.
If it’s too tangy, add a pinch more sugar. If it needs brightness, add a splash of balsamic.
Thicken if needed. If the sauce is thin, uncover and cook on High for 15–20 minutes. Alternatively, stir in a tablespoon of tomato paste or a small knob of butter for body.
Serve. Spoon over pasta, creamy polenta, or rice.
Garnish with fresh basil or parsley and a generous shower of Parmesan.