To make the filling, place the chicken in a medium saucepan and fill with water until the chicken is submerged by about an inch. Add 1 teaspoon of the salt and bring to a boil over high heat. When the water starts to boil, turn off the heat and cover. Let the chicken stand for 15 minutes. Test by cutting into a piece: it should not be pink. Let cool and shred the meat along the grain into tiny shards with your fingers, or chop into a confetti-sized dice. Reserve the stock for another use or discard.
In a small skillet, heat the 1 tablespoon oil over medium-high heat until it becomes runny and starts to shimmer. Add the onion and garlic and cook until the onion is soft and light golden, four to five minutes. Add the chicken, carrots, and green beans, and stir to mix. Add the soy sauce, remaining salt, and 1 teaspoon pepper (or to taste) and mix thoroughly. Add the Chinese chives and stir and cook until the mixture is heated through.
Allow the filling to cool completely.
To assemble the egg rolls, carefully peel one wrapper from the stack (cover the remaining wrappers with a damp cloth to keep them moist). Lay the wrapper on a dry work surface with one corner pointing toward you.) Place 2 tablespoons of filling just below the center line of the wrapper parallel to your body. Shape it into a mound 1 by 3 inches, leaving about 2-1/2 inches on either side.
Fold the corner closest to you over the filling and tuck it under snugly. Roll once, then fold the left and right sides in to form an envelope. Continue to roll the filling tightly into a fat tube until you reach the end of the wrapper. Before you reach the end, dab some egg white or water along the top edge to seal the egg roll. The egg roll should measure 4 to 5 inches in length and 1 to 1-1/2 inches in diameter.
Place on a plate or tray and cover with a damp towel or plastic wrap. Repeat with the remaining filling and wrappers.
Preheat the oven to 250 degrees F. Line a plate with paper towels. In a large wok, heavy skillet, or Dutch oven, heat the 3 cups oil over high heat until it reaches 350 degrees F on a deep-fry thermometer.
Reduce the heat to medium-high. Using tongs, gently lower the egg rolls into the oil one by one; fry in a batch of five or six until both sides are evenly golden brown, one to two minutes. Remove the egg rolls with a slotted spoon, shaking off excess oil, and drain on paper towels. Keep warm in the oven.