Preheat the broiler if necessary. Prepare kebabs by threading turkey, onion, turkey, mushroom, turkey, tomato, turkey, and onion onto skewers. Place kebabs on a broiler pan. Measure 2 teaspoons of olive oil into a small dish or ramekin. Use a brush to sparingly coat all sides of the kebabs with the oil.

In a small dish or ramekin, combine oregano, cumin, 1/8 teaspoon cayenne pepper, 1/8 teaspoon salt, and pepper. Mix well and use your fingers to drizzle spices onto all sides of the kebabs. Broil for 14 minutes, turning once after 7 minutes.

While kebabs are cooking, make the lemon mint sauce in a food processor. Place parsley, mint, and garlic in the work bowl and pulse a few times until items are finely chopped. Add in the lemon zest, 1/8 teaspoon cayenne pepper, and 1/8 teaspoon salt. In a measuring cup, combine the chicken broth, lemon juice, and remaining 4 teaspoons of olive oil. Run the food processor and slowly pour all of the liquid in. Promptly turn off the machine.

When kebabs are cooked, arrange them on a serving platter and pour the lemon mint sauce over them.