Place rice in a medium bowl. Add cold water to cover rice; using your hands, swish and agitate rice in water for 10 to 15 seconds. Pour off water and repeat washing process until water is mostly clear, 4 to 5 times. Pour off water. Add cold water to cover rice by at least 1 inch; let soak, uncovered, at room temperature 30 minutes. (Rice grains should become plump and opaque.) Drain rice in a fine wire-mesh strainer and rinse under cold water for 10 to 15 seconds. Let rice drain completely, about 30 seconds. Set aside.

While rice soaks, grind saffron threads into a fine powder using a mortar and pestle. Stir together saffron powder, lime juice, and 2 tablespoons water in a small bowl until combined. Set aside.

Heat oil and butter in a 4-quart saucepan over medium until butter has melted. Add shallot and 1/2 teaspoon salt; cook, stirring occasionally, until shallot has softened, about 4 minutes. Add garlic; cook, stirring constantly, until fragrant, about 30 seconds. Stir in drained rice; cook, stirring constantly, until grains are evenly coated in oil, about 1 minute.

Add stock, saffron mixture, and 1/2 teaspoon salt to rice mixture; gently shake saucepan to ensure rice is evenly submerged. Bring to a simmer, uncovered, over medium-high. Cover with a tight-fitting lid and reduce heat to low. Cook, covered, 12 minutes. Do not remove lid. Remove from heat; let steam, covered, 5 minutes. Uncover and fluff rice using a fork. Fold in cherries, pistachios, cilantro, and remaining 1/2 teaspoon salt. Transfer to a bowl and serve with lime wedges.