Marinate chicken tenders in 1/2 cup yogurt, pinch of salt, and Sriracha for 1-2 hours.
Meanwhile, prepare tzatziki sauce by mixing remaining 1/2 cup yogurt, dill, grated cucumber, lemon juice, and a pinch of salt and pepper. Set aside.
Preheat oiled grill pan over medium high heat. Grill pan should be slightly smoking. Preheat oven to 300-degrees F.
Remove chicken tenders from marinade and blot away excess marinade. Lay tenders on grill pan and grill 4-5 minutes per side. Remove and set on cutting board. Turn off grill.
Place flatbreads on the residually heated grill until warmed, about 2 minutes.
Slice chicken tenders, set out toppings, and begin to build each gyro on flatbreads with one quarter of the amount for each of the following ingredients: tzatziki, grilled, chicken, red onion, tomato, and sliced cucumber.