1. To prepare cucumbers, place cucumber slices in a colander; sprinkle with salt, tossing well. Drain 1 hour. Place cucumber slices on several layers of paper towels; cover with additional paper towels. Let stand 5 minutes, pressing down occasionally. Combine cucumber, shallots, and next 6 ingredients (through chile) in a large bowl; toss gently. Cover and set aside.
2. To prepare chicken, place each chicken thigh between 2 sheets of heavy-duty plastic wrap; pound to 1/2-inch thickness using a meat mallet or small heavy skillet. Combine soy sauce and next 5 ingredients (through garlic) in a large zip-top plastic bag. Add chicken to soy sauce mixture in bag; seal. Marinate in refrigerator 30 minutes, turning bag occasionally.
3. Heat a grill pan over medium-high heat. Coat pan with cooking spray. Remove chicken from bag; discard marinade. Place 4 thighs in pan; cook 6 minutes on each side or until done. Repeat procedure with remaining 4 thighs. Place 2 thighs and 1/2 cup cucumbers on each of 4 plates; sprinkle each serving with 1 tablespoon green onions and 1 teaspoon sesame seeds.
Wine note: Serve an herbaceous sauvignon blanc to echo the crisp freshness of the dish's cucumber and mint. Chilean versions, like Veramonte Sauvignon Blanc Reserva 2007 ($11), strike a balance between the crisp, green flavors of the New Zealand–style wine and the juicy tropical fruit and melon of California versions in an affordable package. —Jeffery Lindenmuth