Combine mayo, anchovies, parsley, lemon juice, Parmesan, clove grated garlic, and Dijon mustard in a food processor or personal blender. Process until smooth and creamy, scraping down the sides.
Bring a large pot of water to a rapid boil, then season generously with a few tablespoons of Kosher salt. Add dried farfalle, then cook according to package directions until al dente.
In a large bowl, combine cooked, drained pasta, thinly sliced kale leaves, pulled chicken, shaved Parmesan cheese, the Caesar dressing, and freshly ground black pepper. Stir very well to evenly coat. Taste, then adjust seasoning as needed.