Slice chicken breasts into 4-6 thin cutlets. Season with garlic powder, salt, and pepper. Whisk egg in a shallow bowl, dip cutlets in egg, and coat with breadcrumbs.

Heat vegetable oil in a skillet until it reaches 350°F. Fry breaded chicken cutlets for about 5 minutes on each side until golden brown. Drain excess oil.

Mix melted butter, minced garlic, and a pinch of salt for garlic butter. Slice hoagies or baguettes in half, spread garlic butter on cut sides, and toast in the oven at 450°F for 3-4 minutes.

In a bowl, whisk mayo, sour cream or Greek yogurt, Parmesan cheese, lemon juice, Dijon mustard, anchovy paste, Worcestershire sauce, and grated garlic until smooth.

Assemble sandwiches by layering crispy chicken on toasted bread, topping with romaine lettuce and Caesar dressing, and sprinkling additional Parmesan. Close sandwich and serve immediately.