In a medium skillet cook mushrooms and sweet pepper in hot oil over medium heat for 5 minutes or until tender, stirring occasionally. Add garlic; cook and stir for 30 seconds. Remove from heat.
In a large bowl combine egg, mushroom mixture, 1/4 teaspoon of the black pepper, and the salt. Add ground turkey; mix well. Shape into four 3/4-inch-thick patties.
For a charcoal or gas grill, grill patties and onion slices on greased rack of a covered grill directly over medium heat. Grill patties for 10 to 15 minutes or until done (165 degrees F), turning halfway through cooking. Grill onion slices for 8 to 10 minutes or until tender and lightly charred.
Meanwhile, in a food processor combine tomatoes, olives, cheese, mint, lemon juice, and the remaining 1/4 teaspoon black pepper. Cover and process until smooth.
To serve, place spinach on roll bottoms. Top with patties, tomato mixture, onion slices, cucumber slices, and roll tops.