The night before, place the chickpeas in a large bowl, add water to cover by 1 inch and stir in the baking soda. Let stand overnight.
Drain and rinse the chickpeas. Place them in a large pot and cover with fresh water by about 1 inch. Bring to a boil over medium heat, then cover and reduce the heat to low. Cook, stirring occasionally, until the chickpeas are tender, about 1 hour.
Meanwhile, place the poultry and 4 cups of water in a large pot and place over medium heat. As the water is about to boil, skim the froth from the surface. Add the cinnamon sticks and season with salt. Cover the pot and let simmer, stirring occasionally, until the meat is tender, about 45 minutes.
Using a slotted spoon or spider, transfer the poultry from the broth to a cutting board; let cool briefly. If using hens, separate them into whole leg and breast pieces, then remove and discard their skins. If using chicken thighs and drumsticks, simply remove and discard their skins. Strain the broth into a clean, shallow pot and add back the pieces of meat. Remove the chickpeas with a slotted spoon and add to the meat and broth, as well. (At this point you can set the meat and chickpeas aside until you are ready to serve the fatteh. I often prepare the meat and chickpeas the day before and reheat and assemble the dish just before serving.)
Heat the oven to 425 degrees. Spread the pine nuts on a baking sheet and toast until golden brown, 5 to 6 minutes; transfer the pine nuts to a bowl. Lay the pita disks on the baking sheet and toast in the oven until golden brown. Transfer to a rack and let cool. Meanwhile, combine the yogurt and garlic in a bowl and season with salt.
Break the toasted pita into bite-size pieces and spread over the bottom of a large serving platter. Using a slotted spoon, spread the meat and chickpeas over the bread, then sprinkle 2 or 3 tablespoons poultry broth over the meat and chickpeas. Spread the yogurt mixture over the top, then scatter the sautéed pine nuts all over the yogurt. Serve immediately.
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