Preheat oven to 350 degrees F. Lay the chicken between 2 pieces of plastic wrap. Using the flat side of a meat mallet, gently pound the chicken to 1/4-inch thickness. Take care not to pound too hard because the meat may tear or create holes.
Lay 2 slices of cheese on each breast, followed by 2 slices of ham, and 2 more of cheese; leaving a 1/2-inch margin on all sides to help seal the roll. Tuck in the sides of the breast and roll up tight like a jellyroll. Squeeze the log gently to seal.
Season the flour with salt and pepper; spread out on waxed paper or in a flat dish. Mix the breadcrumbs with thyme, kosher salt, pepper, and oil. The oil will help the crust brown. Beat together the eggs and water, the mixture should be fluid. Lightly dust the chicken with flour, then dip in the egg mixture. Gently coat in the bread crumbs. Carefully transfer the roulades to a baking pan and bake for 20 minutes until browned and cooked through. Cut into pinwheets before serving.