Prepare grill or broiler.
Combine first 6 ingredients, stirring until sugar dissolves; set half of lime mixture aside. Combine other half of lime mixture and steak in a large zip-top plastic bag; seal. Marinate in refrigerator 10 minutes, turning once. Remove steak from bag; discard marinade.
Place steak on grill rack or broiler pan coated with cooking spray; cook 6 minutes on each side or until desired degree of doneness. Let stand 5 minutes. Cut steak diagonally across grain into thin slices.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add onion; sauté 3 minutes. Add tomatoes; sauté 2 minutes. Place onion mixture, lettuce, cucumber, and mint in a large bowl; toss gently to combine. Divide salad evenly among 6 plates. Top each serving with 3 ounces steak; drizzle each serving with 1 tablespoon reserved lime mixture.
The salad gets its heat from 2 tablespoons of chile paste. If you prefer milder food, use half. English cucumbers work well for quick meals because they're virtually seedless.