Place potatoes and salt in a large pot and cover with cold water by 1 inch. Heat over medium high heat until water just begins to boil. Reduce heat to maintain a simmer and cook potatoes until tender but not completely cooked through, about 8 to 10 minutes for medium small potatoes. Drain potatoes in a colander and let cool to the touch. Slice potatoes in half and set aside.
Dice your celery and onions.
Prepare a medium heat either on a charcoal or gas grill. Or heat a large grill pan over medium beat. Brush grates with a thin layer of vegetable oil. Place potatoes on a bowl and pour in a little oil and toss to just coat the potatoes. Place the potatoes, cut side down on grill. Cook until dark grill marks develop, about 3 - 4 minutes. Using tongs, flip potatoes and cook until golden brown and cooked through, another 3 - 4 minutes.
In the meantime make the basil mustard vinaigrette: Place mustard in a medium sized bowl. Add minced garlic and white balsamic vinegar. Whisk this all together. Add the olive oil and whisk well. Or place in a mason style jar and shake vigorously until well blended. Stir in the chopped basil.
Add the diced celery, onions and 1/2 cup basil vinaigrette to the bowl that you tossed the potatoes in the oil. Transfer the cooked potatoes to this bowl and gently toss until mixed. Taste and adjust seasoning with salt and pepper. Place in a serving bowl, sprinkle with Parmesan and serve.
Notes
The remaining basil vinaigrette will keep in the refrigerator for up to a week. Use it on your next tossed salad.