Reduce the heat to medium. Add the red onion and cook, stirring occasionally, until softened and lightly caramelized, 6 to 8 minutes. Add the kale a few handfuls at a time, stirring after each addition so that it starts to wilt, until all of the kale is added, about 1 minute total. Add 1/4 teaspoon kosher salt and 1/4 teaspoon black pepper and stir to combine.