For the chicken: Using a meat mallet, evenly pound the chicken breast halves between 2 sheets of plastic wrap to 1/4-inch thickness. Put the chicken into a heavy resealable bag or 9-by-13-inch baking dish. Add the oil, parsley, shallots and garlic; sprinkle the chicken with salt and pepper and mix in the bag or toss in the baking dish until the chicken is coated in the marinade. Let stand at room temperature for 30 minutes.
Using tongs, remove the chicken from the marinade, allowing the oil to drain. (It's ok if some of the garlic and shallots still cling to the chicken.) Discard the marinade.
Heat 2 tablespoons oil in a large (12- to 14-inch) skillet over medium-high heat until it shimmers. Add the chicken breasts, 2 at a time, being sure to space them out in the skillet. (Avoid over-crowding, or you won't get a good sear.) Sear until light golden on the first side, 3 to 4 minutes. Flip and sear on the other side until the chicken is cooked through, 3 to 4 minutes more. Transfer the chicken to a serving platter and cover with aluminum foil to keep warm. Cook the remaining 2 chicken breasts, adding another tablespoon of oil if needed. Reserve the skillet for the sauce.
For the pan sauce: In a small bowl, combine the tomatoes, basil if using and the garlic; season with salt and pepper and toss to combine. Set aside.
Pour out all but 2 tablespoons oil from the skillet. If the skillet is dry and more oil is needed, add about 2 tablespoons. Over medium heat, add the olives and shallots and cook until softened, about 3 minutes. Add the vinegar and cook, scraping up any browned bits in the skillet. Whisk in the chicken broth and any pan juices the chicken released on the platter and bring to a simmer. Cook until the liquid has reduced by half. Stir in the mustard, add the seasoned tomatoes and cook, stirring, until heated through, about 1 minute; season with salt and pepper if needed.
Spoon the sauce over the chicken on the platter, garnish with chopped herbs if using and serve immediately with the Winter Greens Salad on the side if desired.
Winter Greens Salad:
Put the arugula, frisee and radicchio into a salad bowl and toss to combine. Drizzle the vinegar over the greens and toss until coated; drizzle the oil over the top and toss again. Add the cheese, season with salt and pepper and toss until well combined. Serve immediately.