Preheat grill or grill pan to medium-high heat (about 400°F). In a bowl, mix 2 tbsp olive oil, 2 minced garlic cloves, 2 tbsp lemon juice, salt, and pepper. Add chicken cutlets, coat, and marinate 10 minutes at room temperature.

In a food processor, combine anchovies, egg yolk, Dijon, the grated garlic clove, and 2 tbsp lemon juice. Blend until smooth, then slowly pour in 1/2 cup olive oil while blending until thick, about 1 minute. Stir in 1/4 cup grated Parmesan, salt, and pepper.

Heat a skillet over medium heat with 2 tbsp olive oil. Add bread cubes, stir to coat, and cook until golden and crisp, 5 to 7 minutes. Sprinkle with a pinch of salt and set aside.

Grill chicken cutlets for 4 to 5 minutes per side, until internal temperature reaches 165°F. Rest 5 minutes, then slice across the grain.

Rub the inside of a large bowl with the whole garlic clove. Add romaine and half the dressing, toss to coat. Pile onto plates, top with sliced chicken, croutons, and shaved Parmesan. Serve warm.