Slice chicken breasts into thin cutlets, about 1/2 inch thick. Whisk together egg, a splash of milk, garlic powder, salt, and pepper in one bowl. Heat vegetable oil in skillet. Dip cutlets in egg mixture, then coat with breadcrumbs. Fry chicken for about 5 minutes on each side until golden brown. Transfer to paper towel-lined plate.

While chicken fries, combine softened salted butter, minced garlic, and fresh parsley for garlic butter. Slice hoagie or baguette lengthwise and spread garlic butter mixture on inside. Preheat oven to 450°F, place bread on baking sheet, and toast for about 3–4 minutes until edges are golden.

In a bowl, whisk mayo, sour cream (or Greek yogurt), lemon juice, Dijon mustard, and anchovy paste (if using) until smooth. Taste and adjust seasoning. Toss chopped romaine lettuce with some dressing until lightly coated.

Layer fried chicken cutlets on toasted garlic butter side of bread. Top with dressed romaine salad and sprinkle additional Parmesan cheese. Close sandwich, slice in half, and serve immediately.