For soup, in a large saucepan cook onion in hot oil over medium-low heat for 5 minutes, stirring occasionally. Stir in ginger and garlic; cook for 30 seconds more. Add water, edamame, and rice. Bring to boiling; reduce heat. Simmer, covered, for 10 minutes or until edamame and rice are very tender. Remove from heat. Stir in spinach, vinegar, soy sauce, wasabi paste, and 1/4 teaspoon salt; cool slightly.
Transfer soup to a blender or food processor. Cover and blend or process until very smooth. Transfer soup to a bowl; cover and chill for 4 to 24 hours.
Meanwhile, thaw shrimp, if frozen. Rinse shrimp and pat dry with paper towels. Sprinkle shrimp with 1/4 teaspoon salt and the pepper. Coat an unheated nonstick grill pan with cooking spray. Heat over medium-high heat; add shrimp. Cook for 2 to 3 minutes or until shrimp are opaque, turning once halfway through cooking. Transfer shrimp to an airtight container. Cover and chill for 4 to 24 hours.
To serve, stir soup. Ladle chilled soup into twelve 3- to 5-ounce small decorative glasses or cups. Top each serving of soup with a grilled shrimp. If desired, sprinkle with sesame seeds.