This is based on a recipe from Diane Kochilas' book, Against the Grain, a cookbook featuring good carb Mediterranean recipes. My version differs in that I used a George Foreman grill for cooking the chicken.
Preheat an electric grill for high heat. When hot, brush the cooking surfaces with olive oil or use nonstick cooking spray. Season the chicken breasts with salt and pepper, add to grill and cook until lightly browned, in two batches, if necessary. Remove chicken from grill and set aside.
In a large skillet, add the onions and garlic, and cook on low heat until wilted, about 6 to 7 minutes. Add the tomato puree and broth and bring to a boil.
Transfer the chicken breasts to the skillet. Cover and simmer for 30 minutes, or until the chicken is tender. Remove the chicken breasts to a platter and tent with aluminum foil to keep warm.
Whisk together the vinegar and honey in a small bowl and pour into the pan, stirring. Simmer for 5 to 10 minutes, until the sauce is reduced to about 1/2 cup. Drizzle the sauce over the chicken and garnish with chopped parsley.