In a large saucepan of boiling salted water, cook the potatoes until just tender, about 12 minutes. Drain and let cool. Using the heel of your hand, slightly flatten each potato; be careful to keep the potatoes intact.
Light a grill. In a small bowl, whisk the olive oil with the lemon juice and garlic. Season with salt and pepper. Brush the potatoes generously with the dressing and sprinkle with salt.
Grill the potatoes over a medium-hot fire until crisp and browned, about 3 minutes per side. Serve at once.
Make Ahead: The smashed boiled potatoes can be refrigerated for up to 3 days. Bring the potatoes to room temperature before grilling.