Step 1Preheat oven to 350°F. Lightly coat a 12-cup silicone muffin pan with cooking spray; place on top of a large rimmed baking sheet.
Step 2Heat oil in a large skillet over medium heat. Add mushrooms; cook, stirring occasionally, until browned, 6 to 8 minutes. Add spinach; cook, stirring often, until wilted, about 1 minute. Stir in 1/4 teaspoon salt. Remove from heat and let cool for 5 minutes.
Step 3Transfer the spinach mixture to a food processor; pulse until the vegetables are very finely chopped. Divide the mixture evenly among the prepared muffin cups, about 1 tablespoon each. Wipe the food processor clean.
Step 4Add eggs, cottage cheese, pepper and the remaining 1/4 teaspoon salt to the food processor. Process on medium speed until smooth, about 20 seconds. Add Swiss cheese and pulse a few more times to incorporate. Divide the mixture among the muffin cups, a scant 1/4 cup each.
Step 5Bring 2 cups water to a boil. Transfer the baking sheet to the oven and carefully pour the boiling water onto the baking sheet around the muffin pan. Bake until the eggs are set, about 25 minutes. Let cool for 5 minutes before removing from the pan.
To make ahead
Refrigerate egg bites for up to 3 days. To reheat, wrap loosely in a paper towel and microwave for 20 seconds to warm. Alternatively, wrap individually in plastic wrap and freeze for up to 3 months. To reheat, unwrap egg bite, wrap loosely in a paper towel and microwave until warmed through, about 30 seconds.