Slice the chicken breasts horizontally into thinner cutlets. Whisk together the egg and milk, then add garlic powder, salt, and pepper. Combine breadcrumbs and panko in another bowl. Dip each cutlet into the egg mixture, coat in breadcrumbs, and fry in preheated vegetable oil at 350°F for about 5 minutes until golden brown. Drain on paper towels.

In a mixing bowl, combine melted butter, minced garlic, mayonnaise, chopped parsley, and grated Parmesan. Slice hoagies lengthwise, spread garlic butter inside, and toast in a preheated oven at 450°F or under broiler for 3-4 minutes until golden.

In a bowl, whisk together mayo, sour cream or Greek yogurt, grated Parmesan, lemon juice, Dijon mustard, anchovy paste, Worcestershire sauce, and grated garlic. Toss chopped romaine lettuce with the dressing until evenly coated.

Layer crispy chicken cutlets onto the toasted hoagies. Top with Caesar salad mix and sprinkle more grated Parmesan before closing the sandwich. Serve immediately while warm.