Lay the apricots cut-side-down on the grill. Grill until the apricots have solid grill marks and release themselves easily from the grill, 3 to 4 minutes. If you try to remove them and they stick - let them be - they will release in time.
Transfer the apricots to a platter cut-side-up. Top the apricots with spoonfuls of the goat cheese, sprinkle with pepper and garnish with torn rosemary leaves if using. Drizzle with olive oil and sprinkle with salt if using.
This dish is best made with less ripe apricots. If they are too ripe, they can fall apart when grilling. Like all Tuscan recipes, using the freshest ingredients possible will make the best dish.