Preheat the oven to 325ºF. Bring about 2-3 cups of water to a boil in a small pot or kettle while you prep the egg mixture.
In a medium saucepan, heat the olive oil over medium heat. Add the mushrooms, green bell pepper, and onion, and sauté for a few minutes until the veggies are just softened.
To a blender, add the cottage cheese, jack cheese, and egg whites. Blend until nice & smooth. Add the spices, and pulse a few times just to mix.
Line a 12 cup muffin tin with parchment paper baking cups. Split the chopped pepperoni into two piles.Add a spoonful of sautéed veggies to each cup, and split half of the pepperoni amongst the cups. Pour the egg mixture on top until filled almost to the top. Top with the remaining pepperoni and a sprinkle of Italian seasoning. Add the hot water to a rimmed baking sheet or a large baking dish (large enough that the muffin tin fits inside or on top) to create steam in the oven. Place the filled muffin tin on top then transfer to the oven. Bake for 25 to 35 minutes or until the eggs are set in the middle and lightly golden on top.
Allow the egg bites to cool for about 5 minutes then remove from the muffin tin. Transfer to a sheet of parchment paper on a cooling rack for cooling.You can leave in the paper cups for storing or take them out - they hold up great in the fridge either way! Optional final step: set the oven to "broil" and place the egg bites bottom side up in the oven, on parchment paper, for 5 more minutes to really hold the shape and add some color.