Place tamarind in a small bowl. Add 1/2 cup water. Let stand for 10 minutes. Knead tamarind in water to soften. Pass through a small sieve, discarding solids. Measure 1/4 cup liquid; set aside.
Add to a mortar, one at a time, the cilantro root, salt, chopped fresh chiles, garlic, and shallots, pounding each ingredient with a pestle until smooth before adding the next ingredient.
In a small saucepan, heat 2 tablespoons oil over medium-high heat. Add the cilantro paste mixture, and fry until fragrant, about 2 minutes. Stir in palm sugar and remaining 1/4 cup water; simmer until thick. Add reserved tamarind water and 1/4 cup fish sauce, and continue to simmer until reduced and slightly thickened, 2 to 3 minutes. Remove from heat; set aside.
Meanwhile, place the fish in a shallow baking dish. Drizzle with remaining 2 tablespoons fish sauce. Let stand 10 minutes.
Heat 4 inches oil in a large wok over medium-high heat Line a baking sheet with paper towels; set aside. Add whole dried chiles, and fry until crisp, about 30 seconds. Drain on prepared baking sheet; set aside. Add basil leaves and fry until crisp, about 15 seconds. Drain on prepared baking sheet; set aside.
Carefully add fish to hot oil. Fry until crisp, about 8 minutes. Remove to a paper towel-lined baking sheet, let drain. Transfer to a serving platter. Coat the fish with the reserved sauce. Crush the fried chiles over the fish, and garnish with basil leaves. Serve immediately.