Make a slit in the side of each chile and carefully stuff it with the cheese sticks. If the chiles tear, secure them with a toothpick. Just be sure to remove it before serving the chile.
Place some of the flour on a plate and roll each of the chiles in the flour until it’s fairly evenly covered. Shake off the excess flour.
Separate the eggs, putting the whites in a large bowl and the yolks in a small one. Whisk the egg yolks with the flour, water, and salt until they’re thick and creamy.
Whip the egg whites until they form soft peaks.
Pour the oil in a saute pan to a depth of 1 ½ to 2 inches, or until it looks like it would reach a little more than halfway up the side of the chiles. Heat the oil to 350 to 375 degrees or until a little of the batter quickly browns when dropped in the oil, about 20 seconds.
Quickly and gently fold the yolks into the whites so that the mixture is an even color but the whites are not broken down. Dip the chiles in the batter, covering them completely and removing any excess.
Slide the chiles, one or two at a time, into the hot oil. As they are browning on one side, carefully spoon some of the hot oil over the top to assist in setting the batter. When they’re browned, gently turn them over and brown the other side, turning only once. Remove them from the oil and drain on a paper towel.
To serve, place the rellenos on a plate and top with some of the green chile sauce. Serve immediately.
Sauce
Heat the oil in a saucepan over medium heat and saute the onion and garlic until soft.
Add the chile, tomato, broth, and cumin. Bring to a boil, reduce the heat and simmer the sauce for 10 minutes. Stir in the cornstarch mixture and continue to simmer for an additional 5 to 10 minutes to thicken.
