Grilling lemons not only heats them up, which is nice for spritzing on hot food, but helps loosen up and release their juices. As a bonus prize for your effort, they look cool too. For people who like that charred flavor and a burst of sour, even sucking or chewing on one can be irresistible. If you happen to have Meyer lemons to grill, they are just sweet enough so that the cut surface will actually caramelize a bit. The heat will also make them taste sweeter overall. Yum.
To char lemons, heat a grill to medium-high heat (you should be able to hold your hand about an inch over the cooking grate for 3 to 4 seconds) or heat a frying pan over medium-high heat. Then simply cut the lemons in half crosswise, pop out any visible seeds, and brush the cut sides with oil. Use vegetable, canola, or grape seed oil for as little added flavor as possible. If you're looking for a more prominent flavor, olive oil can bring some heft and a bit of a bitter kick to the party. You can sprinkle the cut sides with fine sea salt, too, if you like.
Set the lemons cut side-down on the hot grill or in the frying pan. Cook, ideally without moving but adjust them if the heat seems uneven, until the lemons are heated through and charred on the cut side, about 3 minutes.
Grilled lemons are particularly tasty alongside grilled salmon, but they also add plenty of citrus freshness to grilled chicken (roast chicken is good too!) and grilled veggies. In short, if you're grilling something that tastes good with a bit of lemon on it, why not throw the lemon on the grill too?