I decided to get fancy and take things a step or two beyond shaking and baking my chicken.
Ham and Cheese Chicken Rolls
3 big chicken breasts
6 slices deli ham
1/2 cup shredded swiss cheese
1/2 cup milk
2/3 cup breadcrumbs
1 tbsp oregano
Prep your dipping station by filling a bowl with milk and another bowl with a mix of breadcrumbs and spices. Use as much seasoning as you want.
Butterfly the chicken breasts (here's an excellent how-to from Good Housekeeping) except keep cutting so you end up with two thin halves, instead of stopping most of the way and then spreading it out. This way you will get two wide, thin pieces of chicken out of each breast. Pound them flat if their thickness is uneven. Now you have six flat pieces of chicken.
Salt and pepper them a little, and then lay a slice of ham on each, sprinkling some cheese on top of the ham. You can use sliced cheese too, but I only had a block of swiss, so I shredded it. Works ok either way, but I suspect the rolling is easier when the cheese is shredded. You need to start at one end and roll the chicken as tightly as you can, keeping the stuff inside. It's not easy, because the ham keeps trying to slide out the other side while you're rolling. My end result wasn't as gorgeous as I'd have liked, but with repeated pokings I finally got a roll of chicken with the ham and cheese mostly inside. Use toothpicks to hold the rolls closed - it won't hold completely but it'll do. Dunk the rolls into the milk and then into the breadcrumbs. Roll them around really well, and use your hands to pat breadcrumbs onto every possible surface.
Put the coated rolls onto a greased or nonstick baking sheet (you want one with a lip, because the cheese will ooze), seam-side down, and bake them at 425 for 30 minutes.
Remember to pull out the toothpicks before you eat the chicken!
Honestly, I think this would have been just as good if I'd butterflied the breasts most of the way and just put the ham and cheese inside like a sandwich. The rolling was a pain and I ended up needing two toothpicks to keep most of them closed. I guess if you get good at it, the end result looks all fancy and awesome, but mine was meh. Next time I try this I'll make them ham and cheese chicken "sandwiches" and see how it goes.