Wash the chicken cutlets and pat dry with paper towels. Trim any fat from each cutlet and cut the cutlets in two at the white vein. Remove the white vein. Then, cover a cutting board with wax paper or plastic wrap. Lay the cutlets out on the paper or wrap. Place on top of the cutlets a large piece of plastic wrap or wax paper. Using a wooden meat mallet, pound the cutlets until they are thin, but not so thin that they form holes or can be seen thru. Take off the paper or wrap. Salt and pepper the cutlets.
Cut the stick of butter in half lengthwise and then in 1/4 inch crosswise cuts, forming cubes of butter. Slice and dice the light green scallion or onion stems. (No white bulb part)
Place a cube of butter and a spoonful of diced scallion stems on the top half of the cutlet. Roll up from the top into a roll and secure with a toothpick. Set the roll on a plate. Continue until all cutlets contain scallion stems and butter, and are rolled up and secured with a toothpick. Leave on plate while preparing the egg bath and breading dishes.
In a large soup bowl or medium bowl, break the two eggs into the bowl and add 2 tbsp of water, salt and pepper. Beat the eggs (or just whites) with a fork until just combined well. In another bowl or on a dinner plate, pour out 2 cups of dry bread crumbs. Butter the sides and bottom of a baking dish large enough to hold 6 - 8 breaded chicken rolls or however many you are making.
Dip each chicken roll into the egg bath and then into the breading making sure the onion doesn't fall out and the chicken roll is nicely covered with bread crumb. As you finish breading each one, place it in the buttered baking dish.
When all are breaded, dot the tops with butter and cover the top of the baking dish tightly with foil.
Bake in a 350ยบ oven for 1 hour. After the first 30 minutes, remove the foil and turn them if bottoms are brown. If not, keep checking and turn when bottoms are brown. Bake without foil for the next 30 minutes. They should look nicely browned and crusty. remove from oven and serve,
If you do not plan to serve the chicken rolls that day, then place the uncooked breaded rolls on a foil lined cookie sheet and freeze them in your freezer. When frozen you can either wrap them tightly in plastic and then foil or place them in a freezer bag. They keep well in the freezer and taste as good as the day you made them when you finally decide to bake them.
