Make the garlic butter. In a small bowl, mix softened butter, minced garlic, parsley, dill, lemon zest, 1/2 teaspoon salt, and a few cracks of pepper until smooth.
Shape and chill the butter. Spoon the butter onto a small piece of parchment and form a log about 1 inch thick.
Wrap tightly and freeze for 15–20 minutes, or refrigerate for at least 45 minutes, until firm.
Pound the chicken. Place each chicken breast between sheets of plastic wrap. Pound gently with a meat mallet or rolling pin until about 1/4 inch thick. Season both sides lightly with salt and pepper.
Fill and roll. Cut the chilled butter into four sticks.
Place one stick near the short end of each chicken piece. Fold the sides over the butter, then roll up tightly from the end with the butter, like a burrito. Make sure seams are sealed to prevent leaks. Use toothpicks if needed.
Chill again. Place the chicken rolls seam-side down on a plate.
Chill in the freezer for 15 minutes (or fridge for 30) to help them hold their shape.
Set up the breading. Put flour in one shallow dish. Beat eggs in a second. In a third, mix panko, paprika, garlic powder, a pinch of salt, pepper, and Parmesan if using.
Bread the chicken. Dredge each roll in flour, shaking off excess.
Dip in egg, then coat thoroughly in the panko mixture. Press crumbs on gently to adhere. Double-dip in egg and crumbs if you want an extra-thick crust.
Preheat the air fryer to 370°F (188°C). Let it heat for 3–5 minutes for even cooking.
Oil and arrange. Lightly spray the air fryer basket. Place the chicken rolls seam-side down with space between them.
Spray the tops lightly with oil.
Air-fry. Cook at 370°F for 18–22 minutes, flipping once halfway and spraying again lightly. They’re done when the coating is golden and crisp and the internal temperature hits 165°F (74°C).
Rest and serve. Let the rolls rest for 5 minutes so the butter settles. Remove toothpicks.
Serve with lemon wedges and a crisp salad or mashed potatoes.