Soy Sake Shrimp with Ginger Aioli

1/2 cup soy sauce
2 green onions, chopped
6 tablespoons olive oil, divided
2 tablespoons unseasoned rice vinegar
2 tablespoons sake or dry Sherry
1 tablespoon golden brown sugar
3 garlic cloves, chopped
24 deveined peeled uncooked large shrimp (about 1 1/2 pounds)
1 cup mayonnaise
1 tablespoon chopped peeled fresh ginger
Cooked white rice

Combine soy sauce, green onions, 4 tablespoons oil, vinegar, sake, brown sugar, and garlic in 13×9x2-inch glass baking dish; whisk marinade to blend. Add shrimp and toss to coat. Refrigerate at least 30 minutes and up to 1 hour, turning shrimp occasionally.

Blend mayonnaise and ginger in food processor until smooth. Transfer ginger aioli to small bowl and refrigerate.

Drain marinade from shrimp into small saucepan; bring to boil. Whisk 2 tablespoons boiled marinade into ginger aioli; reserve remaining boiled marinade.

Heat remaining 2 tablespoons oil in heavy large skillet over medium-high heat. Add shrimp; sauté until just opaque in center, about 4 minutes. Mound rice in center of plates. Arrange shrimp around rice; drizzle with ginger aioli. Serve, passing reserved boiled marinade.