Make the dressing: combine mayonnaise, 1/3 cup Parmesan, lemon juice, buttermilk, chopped garlic, rinsed anchovies, and 1–2 tsp chopped chipotle in adobo in a food processor or blender; process until smooth. Refrigerate until ready to use.
Warm the tortillas: cover each with a damp paper towel and microwave for about 20 seconds to make them pliable.
Assemble the wraps: divide the chopped romaine evenly among the 4 tortillas, then top with sliced tomato, chopped cooked chicken, croutons, and the remaining 1/4 cup grated Parmesan.
Dress and wrap: drizzle about 1 tablespoon of the Caesar dressing over the fillings in each tortilla, fold in the sides and roll up like a burrito, then cut in half to serve.