In a hot pan, cook the lobster shells and peppercorns in olive oil until well browned. Add the onion, carrot, and celery and cook until lightly caramelized.
Add the tomato paste and cook a few more minutes, being careful not to add any more color to the tomato paste. Deglaze with brandy and reduce by 1/2.
Add paprika, cayenne, roux, star anise, lobster stock, bouquet garni, and fennel, and whisk to incorporate the roux. Simmer covered for 1 hour.
Strain the sauce, discarding the vegetables. Place lobster shells in a food processor and grind. Return ground shells to the sauce and cook until reduced to sauce consistency. Finish with cold butter whisked into sauce. When butter is incorporated, strain sauce again.