Bring to a boil a pot of salted water with a touch of vinegar added. Using a spoon, gently lower the eggs into the water to prevent them from cracking. Boil 5 minutes for a medium soft egg (8 minutes for hard-boiled). Remove the eggs and immerse them in ice water to stop the cooking process.
Toast the bread slices under the broiler until they reach the desired shade.
In a bowl, mash the avocado with a fork and blend in the lime juice. Spread the mixture on the toasted bread.
Slice the eggs lengthwise and arrange them carefully on the bread slices.
Arrange the radish slices on the bread. Add small dabs of whole grain mustard to taste.
Garnish with a few chive stems. Add a small pinch of fleur de sel to each slice. Enjoy!