Season the chicken generously with salt and pepper. In a medium pot of simmering water, carefully add the chicken and poach until slightly undercooked, about 8 minutes. Remove using tongs and let rest until cool enough to handle; tear or cut into small pieces. (Reserve the broth for another use if desired.)
Meanwhile, heat a large heavy-bottomed skillet over medium heat. Add the pistachios and almonds and cook, tossing the pan or stirring occasionally, until toasted, about 3 minutes. Transfer to a bowl; set aside.
Combine the saffron and 3⁄4 cup hot water in a small bowl or pot; set aside.
Bring a large pot of water to a boil over high heat. Season with 3 tablespoons kosher salt and add the rice. Cook until al dente, about 15 minutes; drain. Season the rice with more salt to taste and stir with two forks.
Meanwhile, in a large skillet, add 2 tablespoons of the melted butter over medium-high heat. Add the garlic and cook, stirring frequently, until lightly toasted, 1–2 minutes. Add the onion and cook until softened, about 4 minutes. Add the chicken and 1⁄4 cup of the saffron water and cook, stirring occasionally, until most of the liquid has evaporated, 3 minutes more.
Combine the chicken mixture with the rice. Stir in the remaining saffron water, the dried fruit, candied lemon, and black caraway seeds. Set aside.
Preheat the oven to 375°. Brush the bottom and sides of a 12-inch Dutch oven or another squat, ovenproof pot generously with melted butter.
Working one piece at a time, lay the lavash across a baking sheet, then brush one side very generously with melted butter. Drape the lavash into the prepared pot with the buttered side facing the pot and one end of the lavash touching the center of the pot (the rest should reach up the sides with 2–3 inches of overhang around the outside rim). Brush the exposed side of the lavash generously with melted butter. Repeat with more lavash, overlapping the pieces slightly, until the whole pot is covered and all the lavash is buttered on both sides.
Add the rice mixture to the center of the lavash, then cover the rice with the overhang, trimming any extra. Place a final piece of butter-coated lavash over the top if any rice shows through. Cover the pot and bake for 40 minutes. Uncover and bake until the top is crispy and golden, about 10 minutes more.
Remove the pot. Place a large serving plate over the opening and carefully but quickly invert the pot to remove the lavash-covered pilaf. Let stand 10 minutes. Cut off the top and serve.