My local lamb (grass paddock raised) just doesn't have enough fat on it to survive a "rack in the oven" experience. Hogget & mutton is fine. This may be your problem too?
So for lamb I would cleave the rack into individual ribs and BBQ on medium-high till the surface has crisped (typically twice as long as a similar beef cut). All the fat should have run. Most people I know don't like eating visible lamb fat
I lightly coat the cut meat surface with oil just before placing on BBQ to ensure an instant sear. You want the lamb to crisp without drying out
If using a wood fired BBQ drop a bunch of fresh lavender (leaf and flowers) under the grill to slightly smoke the lamb
Treat it like beef steak. Super-hot pan, slap it on, keep pressing it with your finger, when it feels very firm, it's well done; about 5-6 minutes a side in general but it will depend on thickness; the finger test never lies. Let it rest for a few minutes after cooking.
Best Answer
If you mean the little paper frills on the ends of the bones on a Frenched rack of lamb, those are purely decorative.