Bring water to a boil in each of the 3 large pots. Salt the water as it reaches a boil. In another pot of boiling, salted water, add the chicken. Reduce the heat and simmer until cooked through, about 6 to 8 minutes. Chill fully in an ice bath and drain. Slice into thin strips and place in a bowl. Refrigerate until ready to use.
Leaving the shrimp in their shells, drop them into 1 pot of boiling, salted water and cook for about 4 minutes. Remove with a slotted spoon to an ice bath until fully chilled. Peel each shrimp, discarding the shells, and slice in half down the length of the shrimp so that you have 2 flat, identical crescent-shaped pieces. Place in a bowl and refrigerate until ready to use.
Drop the bean thread noodles into the third pot of boiling salted water and cook until done, about 2 to 3 minutes. Drain and cool under cold, running water. When cooled, sprinkle with a little sesame oil, to taste. Toss the noodles until they are all lightly coated with the sesame oil. Place in a bowl and refrigerate until ready to use. When ready to assemble the rolls, line up all of the components?the chicken, shrimp, noodles, lettuce, cucumbers, carrots, mint, and basil, in their own containers?on the work surface or table.
Fill a large, flat bowl with warm water, making sure it is large enough to fit the rice paper wrappers. Lay a clean towel on the workspace in front of you. Carefully dip the round wrapper into the water, just to get it wet all over, trying not to crack it. It will still be slightly hard. Remove it from the water and lay it onto the clean towel, where it will continue to soften.
Layer a small amount of each of the ingredients on the bottom 1/3 of the wrap. Start with a few strips of chicken, then 2 to 3 pieces of shrimp, then some noodles, then some lettuce, then some slices of cucumber, followed by some shreds of carrot, and then a pinch each of mint and basil. Fold up the bottom 1/3, fold in the sides and roll up the rest of the length to make an egg roll shape. Set aside and continue with the rest of the ingredients until all of the rice wrappers are used.