Bring 6 cups of water to a rolling boil in a large pot. Add kosher salt (to taste). While the water heats, rinse the basmati rice under cold running water until the rinse water runs clear.

When the water is boiling, add the rinsed rice and stir. Cook, stirring occasionally, until the rice is al dente, about 6–8 minutes. Drain the rice in a fine-mesh sieve and rinse with cold water to stop the cooking; let drain well.

In a medium bowl, combine 1 1/2 cups of the drained rice with the 3 tablespoons plain yogurt and stir to coat the grains evenly. Set aside.

Heat a 10-inch skillet over medium heat. Add the 3 tablespoons salted butter and 3 tablespoons extra virgin olive oil and melt until the butter is sizzling.

Spread the yogurt-coated rice in an even layer across the bottom of the skillet. Pile the remaining rice on top, mounding it toward the center. Using the handle of a wooden spoon, gently poke 5–6 holes down through the rice to the bottom of the pan to allow steam to escape. There should be visible oil around the edges; add a little more oil if you don't see it bubbling up.

Cook the rice over medium heat, turning the skillet a quarter turn every 5 minutes to promote even browning. After about 15 minutes you should see a golden crust forming at the edges. Reduce the heat to low, cover the skillet with a lid (or foil and lid), and continue to cook for another 15–20 minutes, until the edges and bottom are deeply golden and crisp (about 35–40 minutes total).

While the rice cooks, make the spiced chickpeas. Heat a large, high-sided skillet over medium heat and add the 1/4 cup extra virgin olive oil. Add the drained, patted-dry chickpeas, 2–4 cloves garlic (thinly sliced or smashed), the 2 thinly sliced shallots, 1 teaspoon ground turmeric, 1 teaspoon smoked paprika, 1 pinch crushed red pepper flakes, and kosher salt to taste. Cook, stirring occasionally, until the chickpeas are crisped and the shallots are softened and browned, about 8–10 minutes.

Add the 2 tablespoons raw sesame seeds and the 2 cups roughly torn greens to the chickpeas. Toss and cook 1–2 minutes more, until the sesame seeds are toasted and the greens are wilted. Remove from heat.

When the rice is done, remove the skillet from the heat and run a spatula around the edges to loosen the crust. Place a large platter over the skillet and carefully invert the skillet onto the platter so the crisped tahdig is on top (let rest 1–2 minutes if needed before lifting the skillet).

Serve the rice topped with the spiced golden chickpeas and scatter the 1/2 cup fresh mint or cilantro over the top.