Step 1In a large stock pot, heat the sesame and vegetable oils over medium-high heat until shimmering. Add the daikon, garlic and ginger; lower the heat to medium and saut? until slightly colored and fragrant, 4 to 5 minutes.
Step 2Add 4 cups of water, the chicken broth, green onions, mirin, vinegar and soy sauce and bring to a simmer over high heat. Reduce the heat to medium and simmer for 10 minutes.
Step 3Season the skate wing with 1 teaspoon salt. Add the skate wing and shrimp to the pot and cook just until the shrimp turn opaque and the skate is firm, 2 to 2 1/2 minutes. Remove the shrimp and skate wing to a small bowl and set aside.
Step 4Skim any scum off the top of the broth. Strain the broth into a container and chill over an ice bath, or refrigerate for about 2 hours, until very cold.
Step 5Prepare a large pot of boiling water and cook the buckwheat noodles according to package directions (about 2 minutes) until the noodles are just al dente, springy and chewy but not soft. Drain immediately and rinse with cold water to stop cooking.
Step 6Divide the noodles into four large soup bowls. Pour some of the chilled broth over each pile of noodles. Top each bowl evenly with shrimp and skate wing, daikon and cucumber. Garnish with toasted sesame seeds.