Preheat broiler.
Heat the olive oil in a heavy 10-inch, broiler-safe skillet (cast iron) over medium heat. Add the onion and garlic, cooking until softened but not browned. Add the kale, an additional tablespoon of olive oil, and a couple of pinches of salt. Stir until the kale begins to wilt, then cover, reduce the heat to low, and cook for 5 minutes. Stir occasionally.
In a medium bowl, beat the eggs well with a fork. Then, beat in the shredded cheese and a pinch of salt and a crack of pepper.
When the kale mixture is tender and caramelized, remove it from the skillet and whisk it into the egg mixture. Add 1/2 tablespoon oil to the pan, and pour the mixture back into the skillet over low heat, spreading out the kale evenly. Cook over low heat until the top just barely begins to set.
Cook for 1 to 2 minutes under the broiler, until the top is gently browned. Serve warm or at room temperature.