Cut onion half in half. Place 1 onion quarter in a food processor. Add 2 tablespoons parsley, 1/2 teaspoon black pepper, 1/2 teaspoon salt, red pepper, allspice, and garlic; pulse until ground. Add beef, lamb, and egg; pulse to combine.
Shape beef mixture into 8 patties. Place patties on a jelly-roll pan; broil 6 minutes or until done.
Combine 1 tablespoon olive oil, red wine vinegar, 1 tablespoon lime juice, remaining 2 tablespoons parsley, 1/4 teaspoon black pepper, and 1/8 teaspoon salt.
Toss kale with remaining 1 tablespoon oil and 1 tablespoon lime juice. Slice remaining onion. Divide kale mixture among 4 plates; top with sliced onion, sliced cucumber, and tomato. Arrange 2 patties on each salad; drizzle evenly with vinaigrette.