Preheat the oven or toaster oven to 450 degrees F (230 degrees C). Lightly grease a baking sheet.
Lightly coat both sides of poblano pepper with olive oil; place pepper, opened like a butterfly with cut side up, onto the prepared baking sheet.
Bake in the preheated oven until pepper is tender and skin starts to brown, 20 to 30 minutes.
Whisk eggs and milk together in a bowl; fold in tomato, mushroom, green onion, cilantro, salt, and black pepper.
Melt butter in a skillet over medium heat; cook and stir egg mixture in the melted butter until eggs are set, about 5 minutes. Spoon egg mixture into poblano pepper and top with Cheddar-Monterey Jack cheese.
Bake in the oven until cheese is melted, 3 to 5 minutes.