Prepare the unagi sauce. Combine soy sauce, mirin, and sugar in a medium saucepot. Bring to a boil. Reduce heat to medium-low and simmer for 5 minutes or until slightly thick.
Meanwhile, cook noodles according to directions, until al dente. Drain and immediately submerge in ice-cold water. Drain and set aside.
Preheat broiler to medium-high. Place unagi on a lined baking sheet and broil for 7-10 minutes or until heated and slightly charred. Remove from heat and cut into bite-size pieces.
Toss noodles with cucumber, rice vinegar, and a couple spoonfuls of unagi sauce. Assemble dishes. Place noodles into two bowls. Top with unagi, pickled ginger, nori strips, green onions, and a poached egg. Finish with more unagi sauce. Serve.