Add the lemon juice, Dijon, Worcestershire sauce, anchovy paste, garlic, salt, pepper, and honey to a food processor and blend until combined. With the processor running, slowly drizzle in the olive and avocado oils until fully incorporated.

Transfer ½ cup to a resealable bag for the chicken. Add the grated Parmesan to the remaining dressing and blend until smooth. Transfer the dressing to a serving container and refrigerate for 1 hour (or longer).

Slice the chicken in half lengthwise to create 2 thinner slices. Cover with plastic wrap and use a meat mallet to gently pound it to ½-inch thick. Pat dry and lightly season each side with salt and pepper. Transfer to the bag with the marinade and seal out the air.

Marinate in the refrigerator for 30 minutes to 2 hours; 1 hour is ideal. Let the chicken sit at room temperature for 15–20 minutes before cooking.

Place the chopped lettuce in a large bowl. Cover and refrigerate until ready to use.

Preheat the grill to medium-high heat, about 450°F. Clean and lightly oil the grates. (See notes for stove-top instructions.)

Once fully preheated, lightly remove the chicken and discard excess marinade. Grill for 4–5 minutes on the first side, undisturbed, until grill marks develop. Flip and grill for 3–4 more minutes, or until the internal temperature reaches 165°F. Transfer to a plate and let rest for 5-10 minutes, then cut into strips.

Drizzle the chilled lettuce with ½ cup of the dressing. Add the grated Parmesan and half of the shaved Parmesan. Use salad tongs to toss until well coated. Add half of the croutons and toss to combine. Transfer to a serving bowl or platter, if using.

Top with grilled chicken strips and sprinkle with the remaining croutons and shaved Parmesan. Finish with freshly cracked pepper and serve with the remaining dressing.