Place 2 eggs in a small saucepan and cover with cold water by 1 inch. Bring to a gentle boil over medium-high heat. Once boiling, reduce heat to maintain a simmer and cook for 9 minutes for firm yolks. Remove eggs to an ice bath and cool for 5 minutes, then peel and halve or quarter. Set aside.

Prepare chicken

Pat chicken breasts dry and season both sides with 1 tsp salt and 1 tsp black pepper. Heat 1 tablespoon olive oil in a skillet over medium-high heat. When hot, add chicken breasts and cook 5–7 minutes per side (depending on thickness) until internal temperature reaches 165°F (74°C) and the exterior is golden-brown. Remove to a cutting board and rest 5 minutes, then slice thinly.

Assemble greens

Meanwhile, rinse and dry Romaine lettuce, then chop or tear into bite-sized pieces to yield about 8 cups. Place lettuce in a large bowl. Add 1 cup croutons and 1 cup freshly grated Parmesan, keeping some cheese back for garnish.

Combine salad

Add sliced chicken and halved eggs to the bowl with lettuce. Toss gently when dressing is added so eggs keep their shape.

Caesar Dressing

In a medium bowl, combine 1 cup mayonnaise, 1 tablespoon olive oil, 2 tablespoons fresh lemon juice, 1 teaspoon Worcestershire sauce, 1 teaspoon Dijon mustard, 2 cloves minced garlic, and 2 minced anchovy fillets (or 1 teaspoon anchovy paste). Whisk until smooth.

Stir in 1 tablespoon freshly grated Parmesan, 1 teaspoon salt, and 1/2 teaspoon black pepper. Taste and adjust seasoning—add more lemon for brightness, salt for seasoning, or a bit of water (1–2 tsp) to reach desired consistency.

Pour 1/3 to 1/2 cup dressing over the assembled salad, starting with less and adding more to taste. Gently toss to coat lettuce, chicken, and croutons. Plate the salad, garnish with remaining Parmesan, additional freshly cracked black pepper, and serve immediately.