Season and cook the chicken: Pat the chicken dry. Rub with olive oil, salt, pepper, and garlic powder. Grill over medium-high heat 5–7 minutes per side, or roast at 425°F (220°C) for 18–22 minutes, until the internal temperature reaches 165°F (74°C).
Rest 5–10 minutes, then slice.
Make the dressing: In a bowl, whisk egg yolk (or mayo), Dijon, anchovies, garlic, and lemon juice until smooth. Slowly whisk in olive oil until emulsified. Stir in Parmesan.
Season with salt and plenty of black pepper. If too thick, whisk in a teaspoon of water or more lemon juice.
Prep the greens: Wash and thoroughly dry the romaine. Moisture waters down the dressing, so use a spinner or pat dry with towels.
Toss the salad: In a large bowl, toss romaine with enough dressing to lightly coat.
Add sliced chicken, shaved Parmesan, and capers if using. Toss again gently.
Finish and serve: Top with extra Parmesan, a few cracks of pepper, and your crunchy add-in. Serve with lemon wedges for a bright finish.