Spoon the avocado from the peel, and then mash the avocado with a fork. Spoon the mashed avocado on the toasted bread. Then, place the arugula on top of the mashed avocado.
In a small skillet, over medium heat, melt the butter. Once melted, crack the egg directly into the hot butter and skillet. Cook the egg uncovered for about 2 minutes, allowing the whites to set, and then cover the skillet until the yolk is slightly cooked, or fully cooked (depending on your preference). This should only take about 2-3 minutes.
Sprinkle the egg with salt and pepper, and then place it directly on top of the arugula and avocado toast. The arugula will probably wilt a bit due to the hot egg...it's okay!