Preheat oven to 400 degrees F. Spray a baking sheet with nonstick cooking spray; set aside.
Melt butter in a large skillet over medium-high heat. Add diced onion and celery and cook for 3-5 minutes, or until vegetables begin to soften. Add garlic and cook for an additional minute.
Season with salt and pepper then sprinkle flour on top and stir until combined. Slowly mix in chicken broth and cook for 3-5 minutes, or until thickened. Remove from heat and mix in peas and carrots, chicken, and cheese until well combined; set aside.
In a mixing bowl or stand mixer, combine self rising flour and Greek yogurt until a thick dough forms. Roll dough out onto a lightly floured surface into a large rectangle (approximately 10”x14”) then place on prepared pan.
Spread chicken mixture down the center of the dough, leaving 2-3 inches of dough on each side. Use a pizza cutter to cut 1-inch diagonal strips down each side of the dough. Fold 1-inch strips up and over the filling, alternating sides to create a braid.
Brush egg on top and sides then sprinkle fresh parsley on top. Bake for 22-25 minutes, or until crust is lightly golden brown. Remove from oven, let rest for a few minutes then cut into slices and serve.