Place rice in a large bowl and pour in lukewarm water to cover. Using your hands, agitate rice until water becomes cloudy; drain. Repeat process until water stays clear (4–5 times). Cover rice with 6 cups cold water and add a generous handful of salt (water should taste salty). Let soak uncovered at room temperature at least 1 hour and up to 4 hours.

Meanwhile, pour oil into a medium saucepan to come about 2″ up sides. Add shallots and set over medium heat. Cook, stirring occasionally, until shallots are golden and crisp, 10–12 minutes. Using a slotted spoon, transfer shallots to paper towels and season with salt. Strain oil into a small bowl; let cool.

Toss cilantro, dill, parsley, mint, and tarragon in a medium bowl; set aside 1 cup for serving. Dissolve saffron in 2 Tbsp. hot water in a small bowl.

Bring a large pot of salted water to a boil (use a nonstick pot if you have one; it will help the rice release at the end). Drain rice and add to pot, stirring gently to prevent any grains from sticking to the bottom. Bring to a boil and cook until grains begin to float on the surface and dance (rice should be almost tender but with a firm, opaque center), about 4 minutes. Drain rice and rinse under cold water. Spread out on a rimmed baking sheet. Wash out pot.

Pour 3 Tbsp. shallot oil into clean pot. Swirl in 2 tsp. saffron water; reserve remaining saffron water for serving. Oil mixture should barely coat bottom of pot; add more oil if needed. Season lightly with salt. Lay lavash across pot (it can go up the sides a little, but don’t overlap). Spoon a third of rice over lavash; top with a third of herbs and a third of green garlic. Repeat, alternating layers of rice, herbs, and green garlic to form a mound. Use the handle of a wooden spoon to poke several holes in rice (without hitting bottom of pot) to let steam escape. Wrap lid with a kitchen towel and cover pot. Cook over medium heat until herbs are slightly darkened, 10–15 minutes.

Reduce heat to medium-low, uncover and pour melted butter evenly over rice. Cover and cook until rice is fragrant and fluffy, 20–30 minutes. Remove from heat; let sit 10 minutes. Taste rice and season with more salt if needed.

Scoop 1½ cups rice into reserved saffron water and toss to color rice. Layer saffron rice with rice in pot, reserved herbs, and crispy shallots. Top with rose petals if using. Break up lavash crust into pieces and serve alongside rice.