Preheat a grill and set up for medium direct heat. While the grill is preheating, prepare the dressing for the grilled potato salad.
Add the ingredients for the dressing into a food processor. Pulse on high until blended smooth and set aside.
Slice parboiled potatoes in half. Larger potatoes may need to be sliced into quarters. Place them in a large bowl, toss to coat with olive oil or rendered bacon fat, and season with kosher salt.
Place the potatoes cut side down on the grilling grate directly over the heat. Allow the potatoes to grill for two to three minutes per side, until they are nicely charred.
Remove the potatoes from the grill and place them back in the bowl. While potatoes are still warm, add half of the dressing and fold together. Let sit for 10 minutes to allow the potatoes to absorb the dressing.
Add the crumbled bacon, white parts of the green onions, and celery to the potatoes and gently fold together, adding desired amount of remaining dressing.
Garnish with the sliced green portion of the green onions and serve warm or at room temperature.