Gather all of your ingredients and equipment. Make sure your potatoes are scrubbed clean.
Place 2 medium-sized bowls near your cutting board. Peel and chop your potatoes into even, bite-sized pieces, placing your cut potatoes into one bowl and your peels in the other. Discard the peels.
Once your potatoes are cut, run them through cold water to rinse off any starch or remaining dirt.
Fill a pot with cold water, just enough to cover the potatoes without much room over (you can always pour excess water out, if needed!) Season the water generously with kosher salt (the rule is: 'Salty like the Sea' but not too salty!). Add your potatoes.
Bring your potatoes to a full boil, stirring frequently with a wooden spoon. Keep an eye on your pot, as you don't want your potatoes to overcook.
Potatoes are done when you can easily stick one with the tip of a sharp paring knife, but potatoes do not fall apart, approximately 15 minutes or so. Err on the side of undercooking if you're not sure, as they will continue to cook even after they've been set aside to cool; otherwise, you may end up making mashed potatoes instead!
Pour cooked potatoes through a colander, then spread the potatoes out onto a sheet pan to stop the cooking. Place the sheet pan into the refrigerator or even the freezer to cool more quickly. Note: The colder the potato when you go to mix them, the less likely they will be to fall apart. While your potatoes are cooling, finely chop your hard-boiled eggs.
In a large mixing bowl, prepare the dressing by stirring together the mayonnaise, mustard, chopped egg, and minced onion. Season with freshly grated black pepper, celery salt and celery seed. Mix well, then taste, adjusting accordingly. Since the potatoes were boiled with salt, we don't want to add any additional salt until we've tasted everything altogether.
Add the cooled potatoes to the mixing bowl, and carefully fold the potatoes through the dressing without breaking the potatoes up, running the spatula along the bottom of the bowl to ensure all of the pieces are coated. Once mixed, taste and adjust for salt, if needed.Note: Don't worry if some of the smaller pieces break up during mixing, as this will help make the salad creamier. Cover your potato salad and keep refrigerated until ready to eat. Serve chilled!