In a small bowl, whisk together eggs, milk, and chives, and season with salt and pepper.
In a large nonstick skillet, melt butter. Once pan is completely coated, add the egg mixture. Tilt pan back and forth to make sure it's completely coated, then let cook, 2 minutes. Once you can move egg back and forth, carefully flip and cook 2 minutes more.
Transfer to a plate and top with black beans, avocado, cheddar, and salsa. Roll up into a burrito and serve.