Preheat oven to 350 degrees F.
In a large pot bring 3 cups water to a boil. Place lobsters in head first and tightly cover pot. Cook for 7-9 minutes. Remove lobsters and reserve cooking liquid. Once the lobsters are cool remove the meat from the claws, discarding shells, separate the body from the tail remove the insides of the body, leaving just the outer shell, reserve shell. Using scissors, cut on the outer edges of the underside of the tail and remove the meat, discarding the cut shell. You will be left with an empty tail, reserve this shell. Cleaning lobsters over a bowl to reserve any juices.
In an oven at 350, toast the brioche until golden brown, approximately 7-10 minutes. In a large sauté pan heat butter over medium high. Add apricots and pine nuts, cook, stirring frequently until the apricots have darkened and the pine nuts have toasted to a deep brown color. Add celery, onion, and fennel. Stir to combine. Cook for 3-5 minutes until the celery is slightly softened. Add the red pepper flake, parsley, a few cracks of fresh ground pepper and season lightly with salt. Add the toasted brioche and toss to combine. Strain reserved lobster juices into a measuring cup and add enough of the reserved cooking water to make 3 cups total liquid. Add this liquid to the stuffing mixture and cook over low heat for 3-5 minutes until the bread has evenly absorbed all liquid.
Dice all the lobster meat, except 4 claws. Toss the diced meat in to the stuffing. Position the tail and body together on a sheet tray. Divide stuffing amongst shells and generously fill each.
Place sprigs of rosemary under the sides of each shell to help stabilize. Place tray under medium-high broiler for approximately 7-10 minutes until crisped and golden brown.
Add the reserved claw to each lobster. Cook another 30 seconds to warm through. Serve immediately.
The lobster shells should be filled with warm stuffing, if made ahead, gently reheat with a little water prior to stuffing.
The addition of rosemary is a wonderful touch as it slightly chars and adds a delightful aroma that permeates the shell (and your kitchen) however this is not necessary and bunched tinfoil can equally serve to stabilize the shells.