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Make the curry paste: Bring 1/4 cup of water to a boil in a small saucepan. Add the dried chiles and makrut lime leaves (if using) to a small bowl; pour the boiling water on top and let steep for 10 minutes.
While the mixture steeps, separate the cilantro leaves from the stems; set leaves and any tender stems aside for garnish. Roughly chop cilantro stems and add to a small food processor (or the bowl of a large mortar and pestle) along with the shallots, turmeric, lemongrass, ginger, garlic, curry powder, shrimp paste (if using) and lime zest.
Add the soaked chile mixture (including its liquid) and process (or crush with a pestle) until curry paste is smooth, scraping down sides of bowl as needed, about 10 minutes for either method. Curry paste can be made up to 2 weeks ahead; store in a covered container in the refrigerator.
Prepare the fried noodles: In a medium (2-quart) saucepan, bring 3 cups neutral oil to 325 degrees over medium-high heat. Add 4 ounces noodles and fry, flipping once or twice, until light brown, 30 to 60 seconds. Using a slotted spoon, transfer noodles to a paper towel-lined plate; set aside until ready to serve. Cool, strain and store remaining oil in a covered jar for another use.
Heat a large Dutch oven or pot over medium. Add about 1/4 cup coconut cream, scraped from the top of one of the cans of coconut milk, to the pot and let it melt, about 10 seconds. Add at least half of the curry paste — or more, for a more pungent khao soi — and stir until deeply fragrant, about 30 seconds. Add 1 1/2 cups chicken broth and the remaining coconut milk, and increase heat to medium-high. Bring mixture just to a low boil, about 3 minutes.
Use 1 tablespoon salt to season drumsticks, then add them to the boiling curry mixture. Cover, and let simmer until chicken is cooked through, 20 to 30 minutes. Uncover, lower heat, and add remaining 1/2 cup chicken stock along with 1 cup water. (It should be more like soup than stew.) Season to taste with fish sauce, sugar and salt. Turn heat to very low, and cover to keep warm.
Meanwhile, bring 4 quarts water to a boil in a large pot. Add 2 teaspoons salt and remaining 12 ounces noodles. Cook until al dente. Drain and portion into four bowls. Add 2 drumsticks to each bowl, and about 2 cups curry broth. Garnish with fried noodles, pickled mustard greens, sliced shallots, chile paste or oil, reserved cilantro leaves and lime wedges. Serve immediately.