Preheat the oven to 350 degrees F (175 degrees C). Grease a sheet pan with cooking spray.

Heat olive oil over medium heat in a large skillet. Add chicken and cook until lightly browned, 3 to 4 minutes per side. Add onions, allspice, ginger, salt, and pepper. Cook until onions are soft and translucent and chicken is no longer pink in the center and the juices run clear, 5 to 7 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Add toasted almonds and mix well.

Place 1/4 cup chicken mixture inside of each pita bread round; pinch each sandwich closed.

Arrange sandwiches next to each other in the greased pan. Spray tops of each with more cooking spray.

Bake in the preheated oven until toasted, about 5 minutes.