Place noodles in a 9-inch square pan; discard seasoning packet.
Bake at 375° for 7 minutes or until browned, stirring twice. Cool.
Combine vegetable oil and next 3 ingredients, stirring with a wire whisk until blended.
Combine half of noodles, mushrooms, and next 5 ingredients. Add oil mixture; toss gently to coat.
Place radicchio on individual serving plates; spoon vegetable mixture evenly on radicchio. Sprinkle with remaining noodles and sesame seeds.
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