Position racks in upper and lower thirds of the oven; preheat to 425 degrees F. Coat a baking sheet with cooking spray.
Combine turkey, spinach, olives, lemon zest, garlic powder, oregano, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl; mix well. Form into four 4-inch patties. Place on the prepared baking sheet.
Brush oil all over the portobellos, then sprinkle with the remaining 1/4 teaspoon each salt and pepper. Place the mushrooms on a wire rack placed over another large rimmed baking sheet. Place both baking sheets in the oven and bake, switching positions halfway through, until the mushrooms are soft and the burgers are cooked through, 14 to 16 minutes.
Place each turkey patty on a mushroom cap. Top each with 1 tablespoon tzatziki, 1/4 cup arugula and 2 onion rings. Top with the remaining mushrooms.
Tip: To prepare portobello mushroom caps, gently twist off the stems of whole portobellos. Using a spoon, scrape off the brown gills from the underside of the mushroom caps. If you prefer, purchase portobello mushroom caps, rather than whole mushrooms.
