Make the Carrot Gnocchi. In a saucepan, bring carrot juice, butter, and salt to a gentle simmer. Add flour all at once and stir over low heat until a smooth dough forms and the raw flour taste cooks out, about 1 minute. Transfer to a stand mixer fitted with a paddle attachment. Add eggs one at a time, mixing fully between each addition until the dough becomes smooth and glossy. Transfer dough to a piping bag. Pipe into lightly simmering salted water, cutting into bite-sized pieces. Cook until set, then transfer to an oiled tray and chill completely.
Cook the Short Ribs. Heat a pan over high heat and sear short ribs until deeply browned. Add a splash of marinade and cook until reduced and glossy, coating the meat. Remove from heat and slice.
Prepare the Gochujang Sauce & Sear Gnocchi. In a large pan, heat oil over medium heat. Add leeks and carrots with a pinch of salt and sweat until softened. Increase heat, add chilled gnocchi, and sear until lightly golden. Stir in gochujang, then add carrot juice and beef stock. Simmer until the sauce thickens and coats the gnocchi. Season to taste and adjust spice with gochugaru if needed.
Assemble and Serve. Transfer gnocchi to a serving dish. Top with sliced short rib and shredded cheese. Torch or broil until melted and bubbling. Serve immediately.
