Cook the eggs in 1 tablespoon butter or any oil, stirring a lot for fine clumps, infrequently for large. (My first try I went with a green olive oil, which tasted good but tinted the eggs chartreuse!)
Make them as soft or set as you prefer. If you’re in a hurry for full chill, transfer them to a small bowl to cool in the freezer for a few minutes.
I added a sprinkling of chopped celery leaves, 1 tablespoon mayonnaise, 1 teaspoon Dijon, and some sea salt. Stir to combine, breaking the eggs up a little or a lot. Serve cold or immediately at room temperature.