Pat chicken dry and season with salt and pepper. Preheat the oven to 375 F.

Preheat a cast iron pan over medium  high heat and add the avocado oil.

When shimmering, add the chicken breasts to the pan (do not crowd). Cook 2 minutes; then flip. Transfer to the preheated oven and cook to internal temperature of 165F.

Meanwhile, make the dressing. Mix the mayo, half the cheese, mustard, anchovy paste, black pepper and lemon juice in a small bowl.

Rinse the Romaine leaves, pat dry and chop. Add chopped Romaine to a large bowl, drizzle with half the dressing. Stir to coat well.

Slice cooled cooked chicken on the bias and add to salad. Add remaining cheese. Toss to coat.

Serve with reserved dressing.