Place the rice in a large bowl and rinse under cool running water, swishing with your hand, until the water runs clear. Cover with warm water and let soak for at least 30 minutes while you prepare the zirvak.
Pat the lamb completely dry with paper towels. Season lightly with salt.
Set the dry kazan or heavy Dutch oven over high heat. Add the diced tail fat and render it slowly until the cracklings are golden and crisp, about 8 minutes. Lift out the cracklings with a slotted spoon and reserve.
Pour the cottonseed oil into the rendered fat and heat until it shimmers and a faint wisp of smoke rises, about 4 minutes.
Carefully add the reserved lamb bones to the hot fat and fry until deeply browned on all sides, about 6 minutes. Remove and set aside.
Add the lamb chunks in a single layer. Do not stir for the first 2 minutes so a hard sear forms, then turn the pieces and brown them on all sides, about 8 minutes total.
Add the sliced onions and cook, stirring occasionally, until they are deeply golden and any liquid has cooked off, about 10 minutes.
Return the browned bones to the pot. Add the julienned carrots in an even layer over the meat and onions. Do not stir. Let the carrots steam and soften undisturbed for 5 minutes, then gently fold them through twice and cook 5 minutes more, until they slump and turn glossy.
Sprinkle in the crushed cumin seeds, barberries, and whole chiles. Pour in enough hot water to just cover the contents by 1/2 inch. Nestle the whole garlic heads into the surface.
Add 1 tablespoon salt, reduce the heat to a steady gentle simmer, and cook uncovered for 45 minutes so the zirvak builds depth and the lamb becomes tender.
Taste the zirvak broth. It should be assertively salty, like a good seasoned soup. Adjust with more salt if needed; the rice will absorb most of it.
Lift the garlic heads out and set them aside. Drain the rice well and spread it evenly over the zirvak in a smooth layer. Do not stir.
Pour just enough additional hot water down the side of the pot to cover the rice by 1/2 inch. Raise the heat to medium-high and cook, uncovered, until the surface liquid has been absorbed and small craters appear on top of the rice, about 15 minutes.
Using a slotted spoon, gently push the rice from the edges toward the center to form a low dome. Nestle the garlic heads back on top.
Pierce the rice mound straight down to the bottom in 6 or 7 places with a long skewer to create steam vents.
Reduce the heat to the lowest possible setting, cover with a tight lid (wrap the lid in a kitchen towel for an even better seal), and steam undisturbed for 25 minutes.
Turn off the heat and let the plov rest, covered, for 15 minutes.
Uncover, lift out the garlic and reserve. Use a slotted spoon to gently fluff and turn the rice from the bottom, bringing the carrots and lamb up through the grains.
Mound the plov onto a large warmed platter, arrange the lamb pieces and garlic heads on top, scatter over the reserved cracklings, and serve at once for the table to share.