Cut kernels from corn cobs; set kernels aside and reserve cobs.
Thinly slice 1 shallot; set aside. Roughly chop remaining 2 shallots.
Heat oil in a large pot or Dutch oven or pot over medium until simmering. Add sliced shallot and garlic; sauté until starting to toast and become golden brown, 2–5 minutes (shallots will naturally separate in oil). Add red-pepper flakes and sauté to toast, about 10 seconds. Transfer shallot-oil mixture to a small bowl.*
Measure out 1 tablespoon oil from shallot-oil mixture and add back to pot over medium heat. Add chopped shallot, ginger and turmeric and cook, stirring occasionally, until fragrant, 2 minutes; season with salt.
Add potatoes and cook 1 minute. Add reserved corn cobs and broth; bring soup to a boil then stir in corn kernels and coconut milk; bring back to a simmer and cook until potatoes are just tender, 8–10 minutes.
Using tongs, transfer cobs to a plate; let cool.
Meanwhile, transfer half of soup to a blender and blend until smooth; transfer back to pot.*
Scrape back of a knife down length of cooled corn cobs to “milk” cobs; add scraped liquid and solids to soup and discard cobs. Stir in lime juice and season with salt additional to taste.
Top each serving with crispy shallots, garlic, infused oil and fresh cilantro.