For the slaw:
1 bulb fennel, trimmed and very thinly sliced with a sharp knife or on a mandolin
3 to 4 stalks celery with leafy tops, stalks very thinly sliced on an angle and tops coarsely chopped
1 small or ½ medium medium red onion, very thinly sliced
1 lemon, juiced
About 2 tablespoons extra-virgin olive oil (EVOO), for drizzling
Salt and pepper
For the chicken:
2 pounds boneless, skinless chicken breasts or thighs
Salt and pepper
3 tablespoons olive oil
2 cloves garlic, sliced or crushed and chopped
About ⅓ cup balsamic vinegar
3 tablespoons pourable or regular light-brown sugar
2 tablespoons fresh rosemary, finely chopped
About 1 tablespoon Worcestershire sauce
1 rounded tablespoon grainy Dijon mustard
2 cups arugula leaves (a couple of handfuls)
A handful of fresh flat-leaf parsley tops, coarsely chopped
In a large bowl, toss the fennel, celery, onion, lemon juice, and EVOO; season with salt and pepper. Let the slaw marinate while you prepare the chicken.
Heat a large cast iron skillet over medium high. Season the chicken with salt and pepper. Add 2 tablespoons olive oil, two turns of the pan, to the skillet. Add the chicken. Cook until browned and cooked through, 3 to 4 minutes per side. Transfer to a plate. Reduce heat a bit and add the remaining 1 tablespoon olive oil, one turn of pan, to the skillet. Add the garlic and stir until aromatic, about 1 minute. Add the vinegar, sugar, rosemary, and Worcestershire and cook, stirring often, until reduced and thickened a bit, about 1 minute. Stir in the Dijon. Add the chicken and turn until coated. Cook until heated through, about 1 minute more.
Mix the arugula and parsley into the slaw. Serve the chicken whole or sliced with the slaw alongside.