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Place the shrimp heads and shells into a stock pot along with the oil and the daikon and/or onion, if using. Saute over medium high heat, pressing out any tomalley from the shrimp heads with your spatula.
Once the bits that are stuck to the bottom of the pot start to brown and smell wonderfully like grilled shrimp, deglaze with the water and scrape all the bits off the bottom of the pot. Bring to a simmer, then simmer for 5 minutes.
Use a wire skimmer to scoop out the shrimp shells and discard, if bits of veggies can remain in the soup, that's fine. Measure the amount of stock, and you will need to have 4 cups (960 ml) for the next step. If you're short, add water; if you have a little more, that's fine.
Add the stock, lemongrass, galangal, chilies and mushrooms to the pot. Grab the lime leaves and twist them with your hands to bruise, then tear them into chunks, discarding the center ribs, and add to the pot. Bring to a simmer, and simmer for about 5 minutes.
Turn the heat up to bring the soup to a boil. Ladle a little bit of the broth into the Thai chili paste and stir to loosen it so it will dissolve more easily into the soup; pour it into the soup. Add the fish sauce and the shrimp. Cook the shrimp until they're halfway done, 30-50 seconds depending on the size, then turn off the heat and let the residual heat cook the shrimp completely; another minute or so.
Add the lime juice and taste and adjust the seasoning to your liking. It should lead with sour, but have enough sweetness to balance. If it tastes too sour, add the sugar.
Stir in the cilantro leaves and serve with jasmine rice. You can serve the rice on the side, or put the rice in the soup if that's what you prefer.
