Make the compound butter: Combine the soft butter with chopped parsley, grated lemon zest, finely chopped garlic cloves, and season it with salt and pepper. Place in the fridge.
Pound chicken breasts into cutlets: If you have small or medium chicken breasts - place them on a chopping board and cover with plastic foil. Pound them with a meat mallet into thin cutlets. Try to pound them from the center to the left and right side to create cutlets that are more square-shaped than rectangular-shaped. Be careful not to tear the meat. The borders of the cutlet (about 1/2 inch or 1cm) should be pounded very thinly. See photos in the post for a reference.
If your chicken breasts are large it's better to butterfly them: cut the chicken breast horizontally in half but not all the way through, open the breast like a book and pound it as described above.
Wrap the cutlets around butter and shape the cutlets: season the fillets with salt and pepper on both sides, place 1/4 of the chilled butter (shape it roughly into a log) on the right side of each cutlet. Fold the upper and bottom part of the cutlet over the butter, then roll up the cutlet from the right to the left side (see photos in the post for a reference). Form a spindle-shaped log, tightly pressing the edges of the cutlets, making sure the butter is tighly closed in the middle. Spindle shape is the most traditional but you could just wrap these cutlets like a burrito or a wrap.
Chill cutlets: Place the cutlets for 10 minutes in the fridge (this will help the cutlets to keep their shape and not fall apart while frying, you can leave them longer in the fridge).
Prepare your breading station: prepare 3 bowls: one with flour, second with beaten eggs, and third with breadcrumbs.
Bread the cutlets: Dredge each cutlet in the flour, then dip in the egg mixture, then coat it in breadcrumbs. Dip the cutlet again in the egg and coat again in breadcrumbs (this is optional but it makes a thicker breading).
Deep-fry the cutlets (see below an alternative preparation method): add 4-6 cups of oil (the amount depends on the size of the pot) into a medium pot and heat to 340°F-350°F (170-180°C). When the oil is hot, fry 2 cutlets at a time, for about 8-9 minutes for smaller cutlets and 10-11 minutes for bigger cutlets. Transfer to a plate lined with parchment paper, check the temperature in the center of the thickest part of each cutlet - it should be 160°F (71°C) and it should come to the safe temperature for poultry while resting - 165°C (74°C). Repeat with the remaining cutlets.
Alternatively, you can just pan-fry the cutlets in a pan until browned on all sides (you will need about 1/4 cup of oil for that). Transfer the cutlets to an oven-safe baking dish and bake at 325°F (160°C) until done or until temperature in the thickest part of each cutlet reaches 160°F (71°C) - it should come to the safe temperature for poultry while resting - 165°C (74°C).
Leave to rest for 5 minutes then serve and enjoy!