Melt butter in a skillet over medium heat.
Add Canadian bacon and sauté for 2 to 3 minutes.
Pour beaten eggs into the skillet and stir the mixture, using a spatula or wooden spoon.
Continue to stir until small curds form and the scramble just sets. Remove from heat, season with salt and pepper and stir in cheese.
Top each avocado halve with the scramble mixture and finish with a generous sprinkle of chives. Serve.