Whisk together the mayonnaise, vinegar, garlic, mustard, and tarragon and salt and pepper, to taste, in a bowl. Cover and refrigerate for at least 30 minutes before serving.
Put the potatoes in a pot of cold water, add 2 tablespoons of salt and cook the over medium heat until tender, about 10 minutes. Drain, let cool and cut each potato lengthwise into 8 slices.
Heat a grill to high. Brush the potatoes with the oil and season with salt and pepper, to taste, and grill until golden brown and cooked through, about 2 to 3 minutes per side.
Transfer to a serving dish and serve with the aioli alongside.
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