1/4 cup Asian fish sauce (preferably from Phu Quoc)
1/4 cup plus 1 teaspoon sugar
2 1/2 teaspoons black pepper
2 teaspoons salt
1 lb shrimp in shell, peeled and deveined
25 (8-inch) square frozen spring roll pastry wrappers made with wheat flour, thawed
1 large egg yolk, lightly beaten
About 6 cups vegetable oil
Make dipping sauce: Stir together sugar and water until sugar is dissolved. Stir in remaining sauce ingredients, then chill, covered, at least 2 hours.
Prepare filling: Put noodles in a large bowl and cover with hot water by several inches. Soak, pulling noodles apart and stirring occasionally, 10 minutes. Drain noodles and cut into 2- to 3-inch pieces (you should have about 3 cups), then transfer to another large bowl.
Put mushrooms in a bowl and cover with hot water by several inches. Soak 15 minutes. Drain and rinse mushrooms thoroughly, then drain again. Trim off and discard any hard parts from mushrooms. Finely chop mushrooms. (You should have about 2 cups.) Add to noodles.
Pulse shallot and garlic in a food processor until finely chopped, then add to noodles along with carrots, chicken, fish sauce, sugar, pepper, and salt. Pulse shrimp in processor until coarsely ground. (Do not overprocess or it will become pasty.) Add shrimp to noodle mixture. Mix with your hands until well combined. Chill filling, covered with plastic wrap, until cold.
Assemble rolls: Line 2 trays with wax paper.
Transfer one fourth of filling to a small bowl and keep remainder chilled, covered. Place 1 wrapper on a work surface, keeping remaining wrappers covered with a clean kitchen towel (to prevent them from drying out). Cut wrapper diagonally in half to form 2 triangles. With long side of 1 triangle nearest you, put 2 tablespoons filling along middle of long edge of triangle, and shape filling into a thin 5-inch log. Fold left and right corners of wrapper over filling, overlapping slightly and aligning bottom edges. (Wrapper will resemble an open envelope.) Dab top corner with yolk, then roll up wrapper away from you into a long thin roll, making sure ends stay tucked inside. Place on a tray, seam side down. Repeat with remaining triangle. Make more rolls in same manner with remaining wrappers and filling, keeping trays of rolls chilled, loosely covered, until ready to fry.
Heat 1 1/2 inches of oil in a 5- to 6-quart pot over moderately high heat until it registers 365°F on thermometer. Fry rolls in batches of 5 or 6, keeping rolls apart during first minute of frying to prevent sticking, until golden brown and cooked through, 4 to 5 minutes. (Return oil to 365°F between batches.) Transfer as fried to a colander lined with paper towels and drain rolls upright 2 to 3 minutes. To eat, wrap hot or warm rolls in lettuce leaves and tuck in mint and cilantro leaves. Serve with dipping sauce.