Preheat oven to 325 degrees. Rinse the chicken well and pat it thoroughly dry. Place the chicken breasts between 2 pieces of waxed paper, and pound on them gently with a rolling pin or a heavy weight until they are about 1/4-inch thick throughout.
In a small bowl, mix together the goat cheese, garlic, and a generous amount of pepper. Stuff the chicken breasts. Lay a slice of prosciutto on top of 1 piece of chicken (if the prosciutto is larger than the chicken breast, trim to fit). Place one-eighth of the goat cheese mixture on the top third of the prosciutto, and roll up the chicken breast to enclose the cheese. Set it, seam side down, on a clean counter or plate, and repeat with the remaining chicken breasts. They will stay rolled quite easily without needing anything to secure them. Place the bread crumbs in a shallow bowl or plate, and season lightly with salt and pepper. Whisk the eggs in a large shallow bowl and set them next to the bread crumbs.
Dip each rolled chicken breast in the bread crumbs to coat it well, then in the beaten egg, then again in the breadcrumbs. Use 1 hand for dipping the chicken in the crumbs, and the other for dipping into the eggs to minimize the amount of crumbs sticking to your fingers. Heat the oil in a large heavy skillet over medium-high heat. When it is hot but not smoking, brown the chicken breasts turning them, frequently, until they are golden on all sides, about 7 minutes total. (Do not overcrowd the pan. Brown the chicken breasts in 2 batches if necessary.) Remove them from the pan and arrange them in a single layer in an 8 by 14-inch enamel or ceramic baking dish. Place the dish in the center of the oven and bake for 30 minutes.
Let cool and refrigerate. Cut into 6 slices.
Whisk together oil, vinegar and mustard. Season to taste.
Toss salad with vinaigrette. Place on plate. Arrange chicken slices on top of salad.