Heat oil in a large stockpot over medium-high heat; add onion and garlic. Cook, stirring occasionally until translucent, about 4 minutes. Add potatoes and chicken stock. Bring to a boil, reduce to a simmer, and cook until potatoes are very soft, about 15 minutes.
Remove soup from heat, and puree with an immersion blender. Return to heat; add kale, and simmer for 5 minutes.
In a medium skillet over medium heat, cook chorizo until the fat renders and the sausage turns golden brown, about 3 to 4 minutes per side. Remove the sausage, and drain on a paper-towel-lined plate. Thinly slice sausage crosswise. Serve soup with a drizzle of olive oil, and garnish with chorizo slices.
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