Preheat grill to medium-high heat. Using kitchen shears, cut the lobster shells lengthwise down the center. Gently pull the meat up and rest it on top of the shell.

Brush lobster meat with 1 tablespoon olive oil and season with 1/2 teaspoon salt and 1/4 teaspoon pepper.

Place lobster tails meat-side down on the grill. Cook for 5-6 minutes, then flip and cook for another 5-6 minutes until meat is opaque and reaches 140°F internal temperature.

While lobster grills, prepare the dressing by whisking together remaining 3 tablespoons olive oil, lemon juice, orange juice, mustard, garlic, remaining salt and pepper in a small bowl.

In a large bowl, combine mixed greens, cherry tomatoes, cucumber, and red onion. Toss with half of the dressing.

Remove lobster from grill and let rest for 2 minutes. Remove meat from shells and slice into bite-sized pieces.