1. Thread apricot halves onto 2 parallel 6- to 12-inch-long slender metal skewers (you will need several pairs).
2. Mix honey, mustard, and tarragon. Brush half of this mixture onto one side of fruit.
3. Place fruit, basted side down, on a barbecue grill over a solid bed of very hot coals or very high heat on a gas grill (you can hold your hand at grill level only 1 to 2 seconds). Cook until lightly browned, about 2 minutes; baste with remaining honey mixture and carefully turn skewers over. Cook to brown other side, about 2 more minutes.
4. Push fruit from skewers onto a platter.