In a medium bowl, combine the chicken, olive oil, lemon juice, Dijon mustard, 1 tablespoon Caesar dressing, garlic powder, black pepper, salt, and anchovy paste if using. Toss until evenly coated.
In a separate bowl, combine the chopped romaine hearts with 1/2 cup of the Caesar dressing and toss lightly until coated.
Warm the tortillas for 10 to 15 seconds in the microwave or a dry skillet until pliable.
Spread about 1 tablespoon of the remaining Caesar dressing down the center of each tortilla.
Divide the dressed romaine evenly among the tortillas.
Sprinkle the romaine with the grated Parmesan cheese, Parmesan shavings, crushed croutons, and freshly cracked black pepper.
Divide the seasoned chicken evenly among the wraps.
Fold in the sides of each tortilla and roll tightly into burrito-style wraps.
Heat a large dry skillet over medium-low heat.
Place the wraps seam-side down in the skillet and cook for about 2 minutes per side, or until lightly golden and crisp.
Slice in half and serve immediately.