Step 1: Take your chicken breasts and slice them horizontally to get 4-6 thin cutlets. In a small bowl, whisk together the egg, a splash of milk, garlic powder, salt, and pepper. In another shallow bowl, combine your breadcrumbs and Panko. Dip each chicken cutlet into the egg mixture, then coat it in the breadcrumb mixture, pressing gently.
Step 2: Heat about an inch of vegetable oil in a shallow pan over medium-high heat. Once shimmering, add your breaded chicken cutlets (in batches if needed). Fry until golden brown and crispy, about 5 minutes per side. Transfer to a paper towel-lined pan to drain.
Step 3: In a small bowl, combine the slightly melted butter, mayonnaise, minced garlic, chopped parsley, and grated Parmesan cheese. Stir until combined into a spread.
Step 4: Split your hoagie rolls lengthwise, leaving them hinged. Spread the garlic butter mixture generously on the inside surfaces of the bread halves. Place the bread under a broiler or in a 450ºF oven for 3-4 minutes, or until golden and toasted. Keep a close eye to prevent burning.
Step 5: In a bowl, whisk together the mayonnaise, sour cream or Greek yogurt, grated Parmesan cheese, lemon juice, Dijon mustard, anchovy paste, Worcestershire sauce, grated garlic, salt, and pepper. Taste and adjust seasoning.
Step 6: In a large bowl, toss your chopped romaine lettuce with about 3-4 tablespoons of the Caesar dressing, coating it but not drowning it.
Step 7: Open up your toasted garlic bread rolls. Layer crispy chicken cutlets onto the bottom half. Pile a generous amount of the dressed romaine salad on top. Sprinkle with extra grated Parmesan cheese if desired. Close and serve immediately.