Cook the noodles in large pot of boiling unsalted water over medium heat until barely tender and still firm. Drain immediately and rinse with cold water until cool. Drain the noodles really well and transfer to a wide bowl; toss with the sesame oil so they don't stick together.
In a small saucepan, heat the peanut oil over medium-low flame. Add the green onions, ginger, garlic, and chili paste. Cook and stir for a minute until soft and fragrant. Mix in the brown sugar, peanut butter, vinegar, soy sauce, and hot water. Stir until the sugar is dissolved and the peanut butter has smoothed out. Toss the noodles with the peanut sauce until well coated. Serve at room temperature or chilled. Garnish with the sesame seeds, cucumber slices, and cilantro.