Preheat the oven to 375°F. Line a baking sheet with parchment paper.
Spread the diced potatoes onto the prepared baking sheet. Drizzle with the olive oil and season generously with salt and pepper. Toss to coat evenly, then roast for 20–25 minutes, stirring once halfway through, until golden brown and crispy.
While the potatoes roast, prepare the chicken. Set up three shallow bowls: one with the flour, one with the beaten eggs, and one with the panko breadcrumbs. Season the flour and breadcrumbs with the onion powder, salt, and pepper.
Season the chicken cutlets with salt and pepper. Dredge each piece first in the flour, then dip into the beaten eggs, and finally press firmly into the panko breadcrumbs until well coated. Let the breaded chicken rest while the oil heats.
Heat the butter and olive oil together in a large, deep-sided skillet over medium heat. Once the butter begins to sizzle, carefully add the chicken. Cook until deeply golden on both sides and the internal temperature reaches 165°F, about 4–5 minutes per side depending on thickness. Transfer to a cutting board and slice into strips.
Make the dressing by combining the mayonnaise, anchovies, dill pickle chips, garlic, lemon juice, Parmesan, black pepper, and a pinch of salt in a food processor. Blend until smooth. Add a splash of milk if you'd like a thinner consistency.
Arrange the chopped lettuce on a large serving platter. Scatter the crispy potato croutons over the lettuce, followed by the sliced crispy chicken.
Drizzle generously with the Dill Pickle Caesar Dressing, sprinkle with additional Parmesan cheese, and serve with fresh lemon wedges.