Put the potatoes in a pot of salted cold water and bring to a boil. Reduce the heat and simmer until the potatoes are tender, but still firm, 12 to15 minutes.
Drain and let cool before cutting each potato in 1/2-inch-thick slices.
While the potatoes are cooking, combine the vinegar, garlic, mustard, basil and 1/2 cup of the oil in a blender and blend until smooth. Season with salt and pepper.
Heat your grill to medium.
Brush the potato slices on both sides with the remaining 1/4 cup oil and season with salt and pepper.
Place the potatoes on the grill, close the cover, and grill until lightly golden brown and just cooked through, about 2 minutes per side.
Carefully transfer the potatoes to a flat surface. Make stacks by layering the following: 1 slice potato, 1 tablespoon goat cheese, 1 slice potato, 1 tablespoon goat cheese, 1 slice potato.
Place the stacks back on the grill, close the cover, and grill until the cheese begins to melt, about 1 minute.
Transfer the potatoes to a platter, drizzle some of the vinaigrette on and around the potatoes, and sprinkle with the chives.
