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In a medium saucepan, cover sunchokes with 1 inch cold water. Season generously with salt. Set over high heat and bring to a boil, then reduce heat to maintain a gentle simmer. Cook until a paring knife inserted into a sunchoke meets little resistance, about 10 minutes; be careful not to overcook.
Drain sunchokes using fine-mesh strainer or colander. When cool enough to handle, place chokes on work surface or cutting board. Working one sunchoke at a time, use bottom of a heavy skillet to press firmly on each until flattened but still in one piece; take care not to press so hard that sunchokes fall apart.
In a large cast iron skillet, heat oil over medium heat until shimmering. Add sunchokes in a single layer and cook 3 minutes without moving until well browned. Flip chokes, then add butter to pan and allow to melt. Continue cooking 3 minutes longer, spooning butter over sunchokes, until browned on the second side.
Transfer to a serving platter and sprinkle sunchokes with flaky sea salt. Serve immediately.