Preheat the oven. Preheat your oven to 300°F and lightly grease a muffin tin with cooking spray or butter (silicone muffin pans do not require greasing).
Prepare the filling. In a skillet, heat butter over medium heat. Add diced mushrooms and cook until they become tender and release their moisture. Add chopped kale and sauté until wilted. Season with ½ teaspoon of salt, taste, and adjust if necessary before removing from heat and setting aside. The mixture should taste fairly salty.
Prepare the egg mixture. Crack eggs into a mixing bowl and season with remaining ½ teaspoon of salt. Whisk the eggs until well combined.
Fill the molds. Using a measuring teaspoon, add mushrooms and kale mixture to the muffin pan scooping in one teaspoon into each cavity to begin. Evenly distribute any leftover mixture. Pour the egg mixture into the muffin pan, filling each one about ¾ full. Use a chopstick or knife to mix the egg and kale-mushroom mixtures. Top with a few shreds of cheese.
Bake the egg bites. Place the filled molds in the oven. On a different oven rack, add a small baking dish or bowl filled with about two inches of water in the preheated oven. This helps the egg bites retain their moisture. Bake for approximately 15-20 minutes, or until the egg bites are set and slightly golden on top.
Serve and enjoy. Remove the egg bites from the oven and let them cool for a few minutes. Gently remove them from the molds using a knife or spoon. Serve them warm as a delicious breakfast or snack.