Cut green onions into 2-inch-long thin strips.
If frozen, thaw shrimp according to package directions. Peel shrimp; devein, if desired.
Pulse breadcrumbs and next 3 ingredients in a food processor 5 times or until mixture resembles fine crumbs. Transfer mixture to a shallow dish or pie plate.
Dip shrimp in egg; dredge in breadcrumb mixture.
Cook shrimp in hot oil in a large nonstick skillet over medium-high heat 3 minutes on each side or until golden.
Gently toss together green onions, salad greens, next 4 ingredients, and 2 Tbsp. Basil-Lime Dressing in a large bowl. Arrange salad on 4 individual plates; top each with 5 shrimp. Sprinkle with 1/4 cup peanuts, and drizzle with remaining dressing. Serve immediately.