In a bowl, toss the shrimp with 1 teaspoon of the soy sauce and 1/4 teaspoon each of salt and pepper. In another bowl, toss the pork with the remaining 1 teaspoon of soy sauce and 1/4 teaspoon each of the sesame oil, salt and pepper. Cover and refrigerate the shrimp and pork for 30 minutes.
Meanwhile, in a small bowl, cover the dried shiitake with the hot water. Let the mushrooms stand until softened, about 20 minutes. Discard the mushroom stems and thinly slice the shiitake caps.
In a small nonstick skillet, heat 1 teaspoon of the vegetable oil. Add half of the beaten eggs, spreading them over the bottom of the pan. Cook over low heat until almost set, 30 seconds. Carefully turn the egg and cook until set, about 10 seconds. Transfer to a plate and cut into thin strips.
In a large nonstick skillet, heat 1 tablespoon of the vegetable oil. Add the shrimp and cook over moderately high heat, stirring, until just cooked through, about 1 minute. Using a slotted spoon, transfer the shrimp to a bowl. Add the garlic to the skillet and cook until fragrant, about 20 seconds. Add the pork and cook, breaking up the meat, until no pink remains, 2 minutes. Add the pork to the shrimp.
Heat the remaining 2 teaspoons of vegetable oil in the skillet. Add the carrot, jicama, green beans and shiitake and cook over moderate heat, tossing, until the vegetables are softened, about 2 minutes. Add to the shrimp and pork.
In a large saucepan, heat the remaining 1 quart of vegetable oil to 375°. Add the shallot rings and cook, stirring a few times, until browned and crisp, about 2 minutes. Using a slotted spoon, transfer to paper towels to drain, then add to the spring-roll filling along with the egg strips, cucumber and remaining 1/2 teaspoon of sesame oil. Toss well and season with salt and white pepper.
On a work surface, brush the edge of 1 spring roll wrapper with a little of the beaten eggs. Spread 1/3 cup of the shrimp-and-pork filling on the wrapper. Bring the bottom end of the wrapper over the filling and roll up the wrapper like a cigar, folding in the edges as you roll. Repeat with the remaining wrappers, filling and beaten eggs.
Reheat the vegetable oil to 375°. Working in batches, fry the spring rolls, turning once or twice, until golden brown and crisp, about 2 minutes. Drain on paper towels. Keep the cooked spring rolls warm in a preheated oven. Cut each spring roll in half, arrange on a platter and serve with chile sauce.