Thread apricot halves onto 2 parallel 6- to 12-inch-long slender metal skewers (you will need several pairs).
Mix honey, mustard, and tarragon. Brush half of this mixture onto one side of fruit.
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.
Push fruit from skewers onto a platter.