Blister, peel and clean the chiles. Pour the oil into a large, deep, heavy skillet or pot (a 10- to 12-inch one that is 3 inches deep is ideal), and set over medium high heat. When the oil reaches 360 degrees on a thermometer (it will shimmer on the surface but not be so hot it is smoking), lay in 2 or 3 chiles—whatever fits most comfortably. Turn them regularly until they are evenly blistered all over, about 1 minute. Drain on paper towels and repeat with the remaining chiles. Set the oil aside for frying the stuffed chiles. When the chiles are cool enough to handle, clean them: Rub off all the blistered skin. Cut a slit in one side (start about ½ inch from the top, end about ½ inch from the tip) and carefully use your index finger to dislodge the seeds from the seed pod just below the stem. Being careful not to rip the slit all the way to the point, quickly rinse out the seeds. Drain slit-side down on paper towels.
Stuff the chiles. Divide the shredded cheese into 6 mounds, then press each one between your palms to form a fairly compact cylindrical shape. Slide each cylinder of cheese through the slits to fill the well-drained chiles, overlap the two sides of the slits and “sew” them together with toothpicks or a skewer. For the greatest easy in battering and frying, flatten the chiles slightly, lay them on a parchment-lined baking sheet and freeze for about 1 hour to firm them up.
Make the sauce. In a medium-large (4-quart) saucepan, heat the oil or lard over medium. Add the onion and cook, stirring regularly, until richly golden, about 10 minutes. While the onions are cooking, use a blender or food processor to puree the tomatoes and their juice. When the onions are ready, raise the heat to medium-high and add the pureeed tomatoes. Cook, stirring regularly, until it is the consistency of a thick tomato sauce, about 8 minutes. Stir in the broth (this will give you the consistency of a brothy tomato sauce), then taste and season with salt, usually about 2 teaspoons. Cover and keep warm until you’re ready to serve.
Batter and fry the chiles. Reheat the oil to 360 degrees. Using an electric mixer, beat the egg whites and ½ teaspoon salt until they hold stiff peaks (they should look nearly dry). Fold in the yolks, then the flour. Spread about ¾ cup flour onto a plate. Remove the chiles from the freezer. Roll a chile in the flour, shake off the excess, pick it up by the stem and dip it in the batter, swishing it back and forth to cover it completely. Swiftly pull it straight up out of the batter and gently lay it in the hot oil. Fry 2 or 3 at a time (depending on the size of your skillet or pot), gently basting hot oil over the top to set the batter. When they are richly golden on the bottom, about 4 minutes, use a small metal spatula underneath and another (or a spoon) on top to gently turn them over. In 3 or 4 minutes, they’ll be golden underneath and ready to remove with the metal spatula to drain on paper towels. Fry the remaining chiles in the same manner.
As the rest of the chiles are being fried, you can keep the finished ones warm in a low oven. My favorite way to serve them, however, is to transfer them to a parchment-lined baking sheet (I carefully roll them onto one hand, then lay them on the baking sheet) and bake them in a 400-degree oven for 10 minutes or so, to render some of the absorbed oil and crisp lightly.
Serve. Check the warm sauce: it should be the consistency of a brothy tomato soup. Ladle a portion onto deep serving plates (pasta bowls work well here), use a metal spatula to transfer a chile on top and decorate with watercress or parsley if you have it.