Wash the kale and remove the tough stems. Then slice crosswise into thick slices. I usually cut the long slices in half. (I shake the leaves dry but leave some water clinging to them rather than spinning dry. The wet leaves cook more easily. Also, you may prep the kale to this point, and then store in a plastic bag in the fridge until ready to use.)
In a large, oiled skillet (I use my 12-inch cast iron skillet), sauté the onion over medium heat for 2-3 minutes, or until beginning to soften. Add the mushrooms, and cook for 5-7 minutes more, stirring occasionally, or until the mushrooms are golden brown. If your skillet becomes dry, add a drizzle of olive oil.
Add the garlic, stir, and season with a little salt and pepper.
In batches, add the kale, tossing gently to rotate the top layer down. As the kale wilts, add more until you have added it all to the pan. Again, if the skillet gets dry, add a bit of oil. (With a seasoned cast iron skillet, I don’t need more than 2-3 tablespoons beyond the initial light oiling of the pan.) At this point, turn the heat down to low and season again with a little salt and pepper. Remove the pan from the heat as soon as the kale is wilted. (If there are a few pieces that still look crisp, that is good. I think greens gets a little stringy when cooked too long.)
While the kale is finishing, cook the number of eggs desired to your liking. I usually do over-easy or sunny side up, but choose your favorite method.
Spoon kale onto plates, top with egg(s), and a grating of your favorite cheese. Serve with a side of toast or crusty bread, if desired.