Wash the chicken and pat dry. Place the chicken breasts skinned side up on a sheet of waxed paper or plastic wrap. Cover them with a second sheet of paper or wrap and, with a flat (not textured) wooden mallet, lightly beat them until flattened to a uniform thinness of less than 1/4 inch. Set aside.
Cut the ham and cheese into thin slices 1/2 inch wide and 2 inches long. Break the egg into a shallow bowl and beat until smooth. Spread the crumbs on a second shallow bowl. Spread the flour on a dinner plate.
Lay the chicken pieces on a flat work surface skinned side down. Lightly spread the inside of each breast with a little mustard (don't use too much or the mustard will overpower the other flavors). Stack 2 slices of cheese and a slice of ham (2 if they are very thin) in the center of each breast. Fold over the small side of the breast, then fold each end up, like an envelope, and finally fold over the large side. Make sure that the filling is completely encased.
Roll the chicken first in the flour, shake off the excess, and dip each breast in the egg, allowing the excess to flow back into the bowl. Lay it in the crumbs. Roll it carefully so that it doesn't open up (crumbs should not get inside the folds or they won't stay closed when it cooks), patting the crumbs into all sides.
When the piece is coated, lay it on a clean, dry plate and repeat until all the pieces are breaded. Set aside for at least half an hour to allow the breading to set. (You may make them several hours or even a day ahead up to this point. Cover and refrigerate, but take them out at least half an hour before cooking.)
Fill a deep Dutch oven, or a deep-fat fryer with enough peanut oil to come halfway up the sides, at least 2 inches deep. Over medium-high heat, bring the oil to 375°F (hot but not smoking). Add the chicken and fry until golden brown, maintaining a temperature of 365°, about 4 to 5 minutes.
Drain well and serve at once with Sherry Sauce if desired.
Combine last 5 ingredients in a large glass measuring cup. Heat to just warm in microwave to help dissolve bouillon.
Melt butter in saucepan on low. Add flour and stir until smooth and bubbly.Add Milk mixture and heat.