Combine turkey, scallions, 5 teaspoons chile paste and sesame oil in a medium bowl; gently knead together. Do not overmix. Form into 4 burgers, about 4 inches wide.
Oil the grill rack. Grill the burgers until an instant-read thermometer inserted in the center registers 165 degrees F, 3 to 4 minutes per side. Let rest for 5 minutes.
Meanwhile, combine mayonnaise and the remaining 3 teaspoons chile paste in a small bowl. Assemble the burgers on toasted buns with 2 teaspoons of the mayonnaise mixture, 3 slices cucumber and 1/4 cup kimchi each.
*Tip: For the korean chile you can find Annie Chun's brand in most large supermarkets or in Asian Markets. To make a substitute, combine 2 tablespoons each white miso and Asian-style chile sauce (such as sriracha) and 2 teaspoons molasses.