For the dressing: Put the mayonnaise, mustard, relish, oil, rice vinegar, shallots, salt, pepper, celery seeds and paprika in a medium bowl. Use a whisk or spatula to combine (see Cook's Note). Chill in the refrigerator.
For the pickled peppers: Add the vinegar, sugar and 1 cup hot water to a mixing bowl or quart container. Stir until the sugar is fully dissolved.
Add the green peppers and red peppers to the pickling brine, making sure they are fully submerged; if not, add more hot water. Cover with plastic wrap and let it sit on the counter for a minimum of 1 hour. Drain well before using; discard the pickling liquid.
For the potato salad: Add the potatoes, salt and 2 gallons water to a large (4-gallon) stockpot. Bring to a boil over high heat and cook until the potatoes are fork-tender but not mushy, 20 to 30 minutes. Drain and place the potatoes in a large bowl; let cool at room temperature for 20 minutes.
Using a spatula, fold in the dressing little by little. As you mix, smash approximately a quarter of the cooked potatoes so there is a mixture of creamy and chunky. Once the dressing is fully incorporated, fold in the pickled peppers, green onions, celery and eggs. Cover with plastic wrap and refrigerate for at least 1 hour before serving.
This potato salad will hold in the refrigerator for 3 to 4 days. If using a stand mixer to make the dressing, use the paddle attachment on a low setting to blend the ingredients together.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.