Slice chicken breasts thinly so they cook quickly and evenly. Season both sides generously with salt and pepper so the meat stays flavorful under the coating.
Dip chicken pieces first into beaten eggs, then coat them in the breadcrumb and Parmesan mixture. Press crumbs lightly so the coating sticks well and fries evenly.
Heat olive oil in a skillet over medium heat and cook chicken until both sides are golden and crisp. Transfer cooked cutlets onto paper towels to remove excess oil.
Blend garlic, yogurt, Parmesan, mustard, vinegar, anchovy paste, lemon juice, salt, and pepper. Slowly add olive oil while blending to create a smooth, creamy dressing.
Add chopped romaine to a bowl, season lightly, then toss with dressing and Parmesan cheese until coated but still crisp. Avoid overdressing so lettuce stays crunchy.
Place crispy chicken on toasted brioche buns, then pile Caesar salad generously on top. Add extra dressing if desired and close with the bun lid before serving.