1While you preheat your grill to medium-high, rinse your mussels, discarding any that are already open. Cut the lemons in thick wedges or in half crosswise and brush the cut sides with oil.
2Scatter the mussels directly on the grill along with the lemons, cut-side down. Close the lid of the grill and cook for 5 minutes.
3Meanwhile, heat the butter and garlic clove in a small saucepan or in a bowl in the microwave until the butter melts.
4Remove the mussels from the grill – they should all be open; discard any that don’t. Transfer to a bowl and drizzle with the garlicky butter; toss to coat. (Alternatively, drizzle with olive oil.) Transfer to shallow bowls, scatter with parsley and serve with crusty bread and grilled lemon.