Heat oil in a large sauté pan with a fitted lid over medium heat. Add shallot, chile, and garlic; cook 2 minutes, until softened. Add turmeric, cumin, salt, and pepper; cook 1 minute, until aromatic. Add rice and stir to coat. Cook 2 minutes, stirring occasionally, until fragrant. Add vegetable broth, reduce heat and simmer, covered, until rice absorbs all liquid and becomes tender, about 15 minutes. Remove from heat and stir in pomegranates, almonds, and herbs.

Combine yogurt, lemon juice, and honey in a small bowl; stir with a whisk. Add 1 to 2 tsp. water to thin out, if desired. Drizzle yogurt mixture over rice.