Place vinegar and basil in a blender with salt and pepper and blend until smooth. With the motor running, slowly add the oil and blend until emulsified, check for seasoning.
Heat grill to high. Brush potato slices on both sides with oil and season with salt and pepper, to taste. Make stacks with the following layers: potato, 1 tablespoon goat cheese, potato, 1 tablespoon goat cheese, ending with potato. Line the grill with foil and grill the stacks, covered, about 1 minute or until cheese starts to melt. When the cheese is almost melted, slide the stacks from the foil to the grate of the grill just to crisp the bottom, about 1 to 2 minutes more. Drizzle the basil vinaigrette on and around the potatoes and top each Napoleon with some of the basis leaves and chives, if using.