First, make the dressing. In a medium bowl, whisk together the mayonnaise, Worcestershire sauce, anchovy paste, lemon juice, Dijon mustard, olive oil, minced garlic, and ⅓ cup (35g) of the grated Parmesan cheese. Season with salt and pepper to taste. Set aside.
Prepare the salad greens. In a large salad bowl, combine the chopped romaine, sliced kale, and sliced fennel. You can wash and dry the greens before or after chopping.
Slice the chicken breasts in half horizontally to create 4 thinner cutlets. Place them between two sheets of plastic wrap and pound to an even ½-inch thickness. Season both sides of each cutlet generously with salt and pepper.
Set up a breading station with three shallow dishes. Place the flour in the first dish. Lightly beat the eggs in the second. In the third, combine the panko breadcrumbs and the remaining ⅔ cup (65g) Parmesan cheese.
Coat each chicken cutlet first in flour, shaking off the excess. Then dip it into the egg, letting any extra drip off. Finally, press the cutlet firmly into the panko mixture, coating both sides completely.
Heat the neutral oil in a large skillet over medium-high heat. When the oil is shimmering (about 350°F / 175°C), carefully add two cutlets. Cook for 2-3 minutes per side, until golden brown and cooked through. Transfer to a wire rack.
Check that the chicken is cooked to a safe internal temperature of 165°F (74°C). Repeat with the remaining two cutlets, adding more oil if the pan looks dry.
Add a few spoonfuls of dressing to the salad greens and toss to coat. Divide the salad among four plates. Slice the warm chicken cutlets and place them on top of the salad. Serve immediately, with extra dressing and Parmesan cheese on the side.