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Put the corn cobs into a large pot with just enough water to cover (break the cobs in half if they don’t fit). Bring the water, over medium-low heat, to a simmer and cook the cobs for 20 minutes. Discard the cobs and reserve 3 cups of the cooking water.

In a medium saucepan, melt the butter over medium heat. Add the onions and cook, stirring occasionally, until softened, 3 to 5 minutes. Add the corn kernels, chicken broth, cream, and reserved corn water. Season with salt and pepper, to taste, and bring to a boil. Lower the heat and let simmer for about 10 minutes. Add the shrimp and simmer until just opaque, 2 to 3 minutes.

Ladle the chowder into soup bowls and garnish with the green onions. Serve hot.