1. Add the corn cobs and chicken stock to a soup pot set over medium heat and gently simmer the stock to infuse it with the corn flavor, 10 to 15 minutes. Add a pinch of sugar if corn is not in season to enhance the corny flavor. Turn off the heat and set aside.
2. Heat the vegetable oil in a large nonstick skillet over medium-high heat. Add the ginger, scallions and garlic and cook until fragrant, stirring frequently, 1 to 2 minutes. Add the shiitakes and cook until softened and just starting to brown around the edges, 4 to 5 minutes. Season them with salt, transfer to a bowl and set aside.
3. Wipe out the skillet, add the butter and place over medium-high heat again. Add the shallots and cook until softened, about 2 minutes. Add the corn kernels and season generously with salt. Cook until the corn begins to release some juices, about 1 minute. Cover the skillet and let the kernels steam until they just start to become tender, 5 to 6 minutes. Add the heavy cream, bring to a simmer, lower the heat to medium-low and let the liquid thicken and reduce by half, 10 to 12 minutes.
4. Pulse half of the cream and corn mixture in a food processor (or blender) and puree until smooth.
5. Remove and discard the cobs from the chicken stock. Add the pureed corn and the remaining corn and cream mixture to the stock, whisking to help incorporate all the ingredients.
6. Bring the soup to a very gentle boil. Beat the eggs in a large bowl or a square plastic storage container. Pour the eggs into the soup in stages, trying to create wide streams, like a waterfall. It should take four or five passes to stream in all the eggs. Serve garnished with the sliced scallion greens.
When blending hot liquids, remove the liquid from the heat and allow to cool for at least 5 minutes. Transfer the liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.