Fill a large pot or cocotte with water, bring to a rapid boil, and salt generously. Stun and drop the lobster head first into the boiling water. Cover and cook for approx. 5 - 8 min. Meanwhile, peel and chop the carrot, shallots, and garlic cloves. Transfer to a bowl and set aside. Transfer the lobster to another pot and cover with cold water to stop the cooking process. Once cool enough to handle, transfer to a cutting board. Remove the tail. Remove the claws. Discard any remains and save the shells in a separate bowl. Remove claw meat. Remove tail meat and set aside. Cut off legs and add tail to saved shells. If it's a female lobster, scrape out the red coral (roe) and set aside.