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Heat oil in a large, nonstick pot or cast iron Dutch oven until shimmering. Add corn kernels, chopped onion, chili powder, 1/2 teaspoon kosher salt, and 1/2 teaspoon black pepper. Cook over medium-high heat, stirring frequently, until onions are soft and corn is lightly charred in spots, about 8-10 minutes.
Add garlic and cook until fragrant, 1 minute. Remove 1-1/2 cups of corn mixture from the pot and set aside. Stir broth and 6 stripped corn cobs into the pot, scraping up the brown bits on the bottom. Bring to a boil, reduce heat, and simmer for 20 minutes.
Use tongs to remove corn cobs from pot and discard.
Stir sour cream and Cotija into the soup. Transfer soup to a blender or food processor (in batches, depending on the size of your unit for safety) to puree until smooth.
Return soup to pot and if needed, gently reheat to desired serving temperature. Stir in half (3/4 cup) of the reserved corn mixture, lime juice, and 2 tablespoons chopped cilantro. Season to taste with salt, pepper, and chili powder.
Toss remaining 3/4 cup corn with lime zest and 2 teaspoons chopped cilantro. Serve soup, topped with a few spoonfuls of the corn mixture, and an extra sprinkling of Cotija.