Step 1Trim the crusts from the loaf of bread, leaving a solid, evenly shaped rectangle. Using a serrated knife, carefully cut the bread in lengthwise slices just as thin as you can, one-fourth- to one-half-inch thick. You should get 6 to 8 slices.
Step 2Trim the rinds from the cheese and cut it into thin slices. Cut the onion in half and then slice it as thinly as possible, using a mandoline or a very sharp chef's knife. Put the onion in a bowl and season it with vinegar, parsley and salt to taste.
Step 3Spread one-fourth to one-third of the cheese over 1 long slice of bread, being careful not to come too close to the edges -- you don't want it oozing into the pan when it melts. Scatter one-fourth to one-third of the onion over that. Cover with another slice of bread and press together firmly. Repeat with remaining bread, cheese and onions.
Step 4Heat 1 tablespoon butter in a large skillet over low heat. Fry 1 sandwich at a time, gently but firmly pressing down with a spatula or a heavy pan. Cook about 5 to 6 minutes on the first side, about 4 on the second. Repeat with remaining sandwiches, adding more butter as necessary.
Step 5Neatly trim the edges of the cooked sandwiches with a serrated knife and then cut each sandwich into thin crosswise strips, about three-fourths-inch wide. Serve hot.