Hard boil the eggs. I like to do this in my Instant Pot - add 1 cup of water to the bowl, then place the trivet in the bowl and add eggs in a single layer. Cook on low pressure for 6 minutes followed by rapid release. Immediately remove to an ice bath and peel while eggs are still warm to the touch. Set aside to cool in the refrigerator.
Lightly butter toast on both sides, then toast in a skillet until lightly golden brown on both sides, about 1-2 minutes per side. Set aside to cool slightly.
Peel, pit, and chop the avocado, then add to a bowl and mash with a fork. Squeeze lime over avocado and mix in. Divide mashed avocado between the 3 slices of toast, then season with flakey sea salt.
Using a course grater, grate 2 hard boiled eggs over each piece of toast. Note: you will probably end up with some "waste" that can't be grated (unless you get your fingers too) - either add that right on top of your toast, save it for a snack, or discard it.
Add chives, red pepper flakes, and hot sauce on top of the grated egg to taste. Serve immediately.