In a large saucepan, cover eggs with water. Bring to a boil over high heat. Reduce heat to simmer and cook for 6 to 8 minutes (cook for 8 minutes if at higher altitude). Remove from heat and leave eggs in hot water for another 3 to 5 minutes. Alternatively, pressure cook eggs in an Instant Pot for 5 to 8 minutes (lower time if eggs are at room temperature, longer time if cold). Drain and place in ice water until cooled. Peel and chop eggs; place in a bowl and refrigerate until ready to use.

In a large saucepan, cover potatoes with water. Bring to a boil, add salt and cook until just fork-tender (do not overcook or your salad will be mushy!). Drain the water from the pan and return pan to the heat. Shake the potatoes constantly over medium heat for about 1 minute or until dry. Let cool enough to handle then peel and cut into bite-size chunks. Transfer to a large bowl and season to taste with salt and pepper.

In a small bowl, whisk together the olive oil and vinegar. Pour over potatoes and toss carefully, making sure not to mash them. Place potatoes in the refrigerator, uncovered, overnight (see Note below).

Take the potatoes out of the refrigerator and add the reserved chopped eggs, cucumbers, radishes and green onion; toss gently. Add enough mayonnaise to moisten the salad and toss well. Add the mustard to taste; toss to coat. Refrigerate until ready to serve, for at least 1 hour if possible to allow the flavors to blend. Adjust seasonings before serving, if necessary, and give it a give it a gentle toss to evenly distribute the dressing.