Heat the EVOO in a small skillet over medium-high heat, then add in the whites of the scallions, ginger, and garlic. Saut until soft and fragrant.
Meanwhile, mix together the green parts of the scallion, water chestnuts, cabbage, chili paste, honey, soy sauce, sesame oil and shredded chicken in a large bowl and reserve.
Peel the cucumbers partially, so it has 4 stripes of skin going down the entire length of the cucumber. Trim the ends of the cucumbers and cut each cucumber into slices 3/4 of an inch thick -- you should have about 20 pieces. Scoop out the center of each slice with a melon baler, leaving a shell of flesh.
Scoop about 1 to 2 tablespoons of the Asian chicken salad into the cucumber cups and garnish the tops with the toasted sesame seeds.