Grease an 8-inch square baking dish with a little olive oil and set aside.
Heat the olive oil in a large skillet on medium heat. Add the thinly sliced kale and cook until just wilted, about 2 to 3 minutes.
Add the onions and mushrooms, and add salt and pepper to taste. Increase the heat to medium high and cook, stirring frequently, until the mushrooms have released their moisture and the onions are translucent, 3 to 5 minutes. Remove from heat and let cool for 5 minutes.
In a medium bowl, whisk together the milk and eggs.
Line the bottom of the casserole pan with one third of the thinly sliced bread. Spread one half of the kale mushroom onion mixture over the bread. Sprinkle one third of the cheddar cheese over the top. Lay down the second third of the bread slices. Top again with the remaining kale mixture, and then with a third of the cheese. Lay the remaining slices of bread over the top.
Pour the milk egg mixture over the bread, pressing down with a spatula so that the milk is absorbed by the casserole layers. Sprinkle the top with the remaining cheddar cheese.
Let the casserole sit for about 15 minutes to further absorb the milk while you preheat the oven to 350°F (180°C). Cover with foil. Bake covered for 30 minutes, then remove the foil and bake an additional 25 to 30 minutes, until the cheese is golden and bubbly. Let the casserole sit for a few minutes before serving.