For the Cucumber-Yogurt Dip: Combine all the ingredients in a small bowl. Cover and refrigerate until ready to use.
For the Spring Rolls: In the bowl of a food processor, pulse together the shallot, garlic, and carrot until finely chopped. Add the mushrooms and pulse until coarsely chopped. Add the shrimp, sugar, soy sauce, salt, and pepper. Pulse until the shrimp is chopped into 1/4-inch pieces.
Place a spring roll wrapper on a work surface. Brush the edges with the beaten egg. Place a heaped tablespoon of the filling about 1-inch from the closest edge of the
wrapper. Spread the filling into a log, approximately 4 1/2-inches long and 1-inch wide. Place a heaped tablespoon of the cooked noodles on top. Fold the closest edge of the wrapper over the filling. Fold the right and left edges of the wrapper over the filling. Roll up the spring roll tightly and place, seam side down, on a parchment paper-lined baking sheet until ready to cook. Repeat with the remaining spring roll wrappers.
In a high-sided saucepan, heat the vegetable oil over medium-high heat until it registers 375 degrees F on a deep-fry thermometer. Cook the spring rolls, in batches, for 2 to 3 minutes, turning occasionally, until golden brown. Drain on paper towels.
Cut the spring rolls in half diagonally and transfer to a serving platter. Serve immediately with the Cucumber-Yogurt Dip.