For the crispy pita: Pour enough vegetable oil in a large heavy-bottomed pot to measure about 1-inch deep and heat to 350 degrees F over medium-high heat. Cut the pita bread into large 1 1/2-inch squares. Fry the bread in the hot oil until golden and crunchy. Transfer to a paper towel-lined plate to drain.
For the chickpea broth: Add 1 can drained chickpeas to a large pot. Cover with 3 cups water, then add the garlic, chile powder, cumin and 1 teaspoon salt. Bring to a boil and turn off the heat.
For the hummus: Combine the 3 cans drained chickpeas in a food processor with the tahini, garlic paste, 2 teaspoons salt and the lemon juice. Blend until just combined. With the food processor on, drizzle in the ice water until blended and smooth. Remove 1 1/2 cups of the hummus to an air-tight container and store in the refrigerator for up to a week. Add the yogurt to the food processor with the remaining hummus and blend until smooth.
To assemble: Put the fried pita in the bottom of a 2-quart serving dish and pour over about 1 cup of the hummus cooking liquid, half of the whole chickpeas, all the yogurt-hummus mixture. Spread it in an even layer, then sprinkle on the remaining whole chickpeas (discard the garlic cloves).
Heat the ghee and olive oil in a medium skillet over medium heat. Add the pine nuts and fry until golden brown, 1 to 2 minutes. Top the hummus with the pine nuts and oil, then sprinkle with about 1/4 cup pomegranate seeds, some parsley leaves and chile pepper.