Preheat your grill to medium heat (about 350°F).
Wash and scrub the potatoes thoroughly, then pat them dry.
Place each potato on a cutting board and make several evenly spaced cuts across the top, slicing down about 3/4 of the way through the potato. Be careful not to cut all the way through.
Brush the potatoes with olive oil, ensuring the oil gets into the cuts. Sprinkle with salt, black pepper, garlic powder, and paprika.
Wrap each potato in aluminum foil and place them on the grill. Cook for about 30-40 minutes, turning occasionally, until the potatoes are tender.
While the potatoes are grilling, prepare the chive butter by mixing the softened butter, chopped chives, lemon juice, and salt in a bowl until well combined.
Once the potatoes are cooked, remove them from the grill and unwrap the foil.
Serve the potatoes hot, topped with a generous dollop of chive butter.