In a medium bowl, combine the mayonnaise, Greek yogurt, lemon juice, garlic, Dijon mustard, and anchovy paste.
Season lightly with black pepper and only a small amount of salt, since the anchovy paste is naturally salty.
Stir in the grated Parmesan cheese.
Slowly drizzle in the olive oil while whisking continuously until the dressing becomes smooth and creamy.
Taste and adjust the seasoning if needed. Cover and refrigerate while preparing the salad.
Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper or a silicone baking mat.
Arrange the chicken breasts on the prepared baking sheet.
Sprinkle both sides with the Italian seasoning, garlic powder, salt, and black pepper.
Roast for 20 to 25 minutes, or until the thickest part of the chicken reaches an internal temperature of 165°F (74°C).
Remove the chicken from the oven and let it rest for 10 minutes before slicing into strips.
Place the chopped romaine lettuce into a large salad bowl.
Add the shaved Parmesan cheese.
Pour about one-third cup of the Caesar dressing over the lettuce and gently toss until evenly coated.
Divide the salad among serving plates.
Arrange the sliced chicken over each salad.
Serve with the remaining dressing and additional shaved Parmesan on the side.