In a medium bowl, whisk eggs to blend with 1 tsp. chile pepper, 1/2 tsp. salt, and black pepper.
Heat a 10-in. cast-iron skillet over medium heat. Add olive oil and 3/4 of green onions and cook, stirring, until softened, 30 seconds. Pour in egg mixture. Cook, stirring occasionally, until softly set, 2 minutes. Reduce heat to low, sprinkle with cheese, and stir. Cook until cheese is softened and eggs are just set, 1 minute.
Scatter eggs with avocado and remaining green onion, chile pepper, and salt.