Trim woody ends from kale stems. Remove ribs and stems from leaves and finely slice crosswise. Coarsely chop leaves; keep them separate from stems.
Heat grapeseed oil in a large skillet over medium. Cook stems, tossing occasionally, until crisp and tender. (About 3 minutes.)
Add leaves a handful at a time, tossing and letting them wilt slightly before adding more. Cook until all of the leaves are wilted. (About 2 minutes.)
Add garlic, perilla oil, soy sauce, and cook, tossing often, until leaves are soft. (About 3 minutes.) Season with salt if needed.