Place the chicken in a saucepan, cover with water, and add the cardamom and 1 tablespoon salt. Bring to a boil, then reduce the heat and simmer until the chicken is cooked through, about 30 minutes. Remove to a plate and set aside or refrigerate if not assembling the wraps immediately.
In a large pan, drizzle the olive oil over medium heat. Add the onions, reduce heat to medium low, and cook, stirring often until softened and starting to caramelize, about 30 minutes. Season with the 1 teaspoon salt, cumin, black pepper, and cinnamon.
Thinly slice the cooked chicken and mix into the onions along with the sumac.
Preheat oven to 350˚F. Lightly grease a baking sheet with olive oil. If using large sheets of flatbread, cut each piece into two triangles.
On a work surface, brush one side of a triangle with olive oil and flip over. Brush with olive oil and sprinkle with sumac. Place a line of about 2 tablespoons of the chicken and onion mixture on the long edge of the bread. Top with a few pine nuts. Wrap up tightly and place on the prepared baking sheet. Repeat with remaining wraps.
Bake in preheated oven until toasted and the edges are crisp, 15-20 minutes.
Serve immediately garnished with more sumac, pine nuts, and a sprinkling of fresh parsley if desired.