Lay out the chicken cutlets with the thin end closest to you. Sprinkle cheese evenly over each breast. Divide the cooled stuffing into 8 equal portions, place one portion on the 1/2 of the cutlet closet to you. Fold the opposite end of the stuffing over, enclosing the cheese and stuffing into a pocket. Use a toothpick to close up the edges and season the outsides with salt and pepper.
Place a large skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Sear the chicken in the hot skillet until golden brown but not cooked through, about 2 minutes per side.
Remove the browned chicken to a plate and melt the remaining tablespoon of butter in the pan. Add the shallots to the butter and cook them until tender, about 2 minutes. Add the white wine and chicken stock to the pan, bring it up to a bubble and return the browned chicken to the skillet. Continue to cook the stuffing pockets until the chicken has cooked through and the sauce has reduced somewhat, 4-5 minutes.
While the chicken is finishing cooking, grab a large salad bowl and toss together the sliced fennel, fennel fronds, sliced apples and arugula. Squeeze the juice from the zested lemon over the salad, then drizzle it with about 1 tablespoon EVOO and season it with salt and pepper. Toss the salad to evenly coat it in dressing and top it off by shaving slices of Parmigiano-Reggiano over the top with a vegetable peeler. Serve each person two chicken roll-ups topped with some of the pan sauce and a side of salad.