1. In a large bowl, combine half each of the vinegar and rosemary. Season with salt and pepper. Add the chicken and toss to coat. Cover and marinate at room temperature for 15 minutes or in the refrigerator for up to 4 hours, turning the chicken occasionally.
2. Place the onions in a large bowl and add the remaining red wine vinegar and a pinch of salt. Mix and set aside for 15 minutes to 1 hour, tossing occasionally.
3. Heat a grill or grill pan to medium-low. Place the chicken thighs on the grill and cook for 15 to 20 minutes, or until nicely browned and cooked through, turning the chicken halfway through grilling. Transfer the chicken to a plate and sprinkle with basil. Set aside.
4. Add the figs, arugula and remaining rosemary to the onions and vinegar. Gently toss the salad, then season with salt and pepper. Divide the salad among 4 serving plates. Place the chicken thigh halves around the salad, then sprinkle the salads with shaved Parmigiano-Reggiano.
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