Combine lemon juice, soy sauce, mustard, garlic, brown sugar, and curry paste in a shallow dish or resealable bag. Add shrimp and turn to coat. Cover or seal and refrigerate for 1 hour.

Preheat an outdoor grill for high heat. Lightly oil the grate. Thread shrimp onto skewers or place in a grill basket. Transfer marinade to a saucepan; bring to a boil and cook for 3 to 5 minutes.

Grill shrimp, turning once and basting occasionally with the boiled marinade, until opaque and cooked through, about 3 minutes per side.