Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender but still holding their shape, about 15 minutes. Drain.
Meanwhile, heat 2 tablespoons olive oil in a large frying pan over medium-high heat until quite hot. Toss in pieces of kale; reduce heat to medium. Cover and cook, stirring often, until kale begins to soften and crisp slightly, about 5 minutes. Add onion; cook until softened and translucent, 2 to 3 minutes. Stir in garlic.
Add remaining olive oil when onions are beginning to brown. Stir in potatoes and season with salt and pepper. Cover and cook until potatoes are completely softened, about 5 minutes. Remove lid; cook and stir until potatoes just begin to brown, about 10 minutes more.
