How to Make It
Using a thin sharp knife, cut the corn kernels from the cobs and transfer them to a bowl. Working over the bowl to catch the liquid, scrape the corn milk from the cobs with the back of a knife.
In a large soup pot, melt the butter in the oil. Add the shallots and cumin seeds and cook over moderate heat, stirring occasionally, until the shallots are translucent, about 5 minutes. Add the corn and cook for 1 minute. Add the vegetable stock and cook for 25 minutes.
Working in batches, puree the soup in a blender until smooth. Pass the soup through a fine sieve, pressing to extract as much pulp as possible. Rinse out the soup pot and return the soup to it. Add the cream and season with salt and white pepper.
In a large skillet, heat the olive oil until shimmering. Add the shiitakes and cook over moderately high heat until softened, about 4 minutes. Add the shallot and garlic, season with salt and pepper and cook for 1 minute. Stir in the stock and cook until the shiitakes are dry, about 3 minutes. Transfer to a plate to cool.
In a bowl, combine the mayonnaise, egg, mustard, cayenne and 1 teaspoon of the chives. Fold in the shiitakes and the crabmeat and then the bread crumbs. Shape rounded tablespoons of the mixture into 18 small crab cakes and transfer to a baking sheet lined with lightly floured wax paper; refrigerate until firm.
Reheat the soup. Meanwhile, dust the crab cakes with flour, tapping off any excess. Melt the butter in a large nonstick skillet. Add the crab cakes and cook over moderately high heat until browned and crisp, about 2 minutes per side. Transfer the crab cakes to a paper-towel-lined plate and season with salt.
Ladle the soup into shallow soup plates and arrange 3 crab cakes in each plate. Sprinkle with the remaining 1 teaspoon chives and serve.