Cut off and discard any dry or wilted bits from the kale and wash the remaining greens in cold water. Transfer to a colander to drain, then coarsely chop the stems and leaves into 2- to 3-inch pieces.
In a large wok or skillet, heat butter and oil over medium-high until rippling. Add scallions and cook, stirring, until softened, about 1 minute. Add kale and cook, stirring, just until wilted, about 1 minute.
Add coconut milk and soy sauce and bring to a simmer. Simmer, uncovered, stirring frequently, until kale is cooked to your taste, about 7 minutes for bright and crisp greens or 10 minutes for darker, softer greens. Season to taste with salt and pepper and serve immediately.