Preheat the oven to 425˚ F. Mix the saffron and 1 tablespoon hot water in a bowl; set aside to steep. Melt 2 tablespoons butter in a medium Dutch oven over medium-high heat. Add the shallots and a pinch of salt; cook, stirring, until golden brown and crisp, 4 to 5 minutes. Remove to a small bowl with a slotted spoon.
Season the chicken on both sides with the cumin and a generous pinch each of salt and pepper. Add to the Dutch oven and cook, turning once, until lightly browned on both sides, 6 to 8 minutes; remove to a plate.
Add the remaining 1 tablespoon butter to the pot along with the ginger and garlic and cook until toasted, about 30 seconds. Stir in the rice until well coated, then pour in the chicken broth, 1/2 cup water, the saffron with its liquid, ½ teaspoon salt and a few grinds of pepper. Bring to a simmer, then stir, lay the chicken on top and cover. Transfer the pot to the oven and cook until the chicken is cooked through and the rice has absorbed the liquid, about 15 minutes.
Transfer the chicken to a cutting board and let rest 5 minutes. Add the peas to the rice; fluff with a fork. Season the yogurt with a pinch each of salt and pepper. Slice the chicken and serve with the rice. Drizzle with the yogurt and top with the shallots.