In a medium saucepan, cover your Jerusalem artichokes with a couple of centimetres of water. Add some salt (the water should taste nicely salted, as if you were seasoning soup). Set over high heat and bring to a boil, then reduce heat to maintain a gentle simmer. Cook until paring knife inserted into a sunchoke meets little resistance, about 10 minutes; be careful not to overcook.
Drain your Jerusalem artichokes using a colander. When they are cool enough to handle, place them on a board. Working 1 Jerusalem artichoke at a time, use the bottom of a heavy skillet to press firmly on each Jerusalem artichoke until it is flattened but still in one piece; take care not to press so hard that the sunchokes break apart.
In a large cast iron skillet, heat oil over medium heat until shimmering. Add Jerusalem artichoke in a single layer and cook without moving until well browned, about 3 minutes. Flip Jerusalem artichokes then add butter to the pan and allow to melt. Add half of thyme to the melted butter and continue to cook, spooning butter over sunchokes, until browned on the second side, about 3 minutes longer.
Transfer Jerusalem artichokes to a serving plate and spoon the thyme butter on top. Garnish with remaining freshly picked thyme leaves and sprinkle with flaky salt. Serve immediately.
