Cool potatoes until they are cool enough to handle, about 30 minutes. Meanwhile, combine olive oil, butter, garlic and rosemary in a small mixing bowl and mix until well combined. Add a couple of cranks of salt and pepper each to the mixture and stir.
Line a baking sheet with foil or parchment paper and gently press each potato with the palm of your hand until split open and flattened to about 1-inch thickness (don't press too flat or potatoes will break apart).
Brush potato tops with butter and oil mixture; top with cranks of salt and pepper.
MAKE AHEAD: This can be made 2 to 3hours ahead. Let stand at room temperature, or the night before, then cover with plastic wrap and store in refrigerator.
Prepare grill to medium-high heat. Place potatoes, oiled side down, on grill. Brush backsides with oil mixture. Cook until crisp and beginning to color, about 5 minutes per side. Serve warm.