Butterfly the chicken: With the breasts facing you lengthwise, cut 3/4 through the center of thicker, long side of each chicken breast and open flat. Pound to an even thickness between two sheets of plastic wrap. Put the chicken in a large shallow baking dish.
Puree the buttermilk, 2 tablespoons lemon juice, 2 tablespoons olive oil, fennel seeds, garlic, basil and 1 teaspoon salt in a blender or mini food processor. Pour over chicken, turning to coat. Cover and marinate 15-30 minutes.
Coat a heavy grill pan with cooking spray or lightly brush with oil and heat over medium high heat until the pan is very hot (a few drops of water should sizzle and instantly evaporate). Cook chicken until it's no longer pink inside, about 3 minutes per side. You might have to do this in batches depending on the size of your grill.
To make the fennel salad:
In medium bowl, mix together remaining lemon juice, olive oil, lemon zest, chili and salt to taste. Add the cannellini beans, sliced fennel, parsley and toss gently.
Slice chicken 1/2-inch thick. Put the chicken and some arugula on a serving plate and spoon fennel salad over the arugula. Use a vegetable peeler or small sharp knife to shave curls of cheese over the plate.
If you don't want to bother with butterflying the chicken breasts, ask your meat counter to do it or buy 8 small chicken cutlets.
Marinate the chicken up to 2 hours ahead. The chicken will take less than 10 minutes to cook.
You can use a stovetop grill pan or fire up your outdoor grill.