1. Prepare tomato sauce: Cook garlic in olive oil until golden. Add basil and onion, cook 2 minutes. Add peeled tomatoes, cook 5 minutes until softened. Strain and set aside.
2. Make lobster stock: Preheat oven to 200°C. Roast lobster shells with onion, celery, and carrot for 10-15 minutes. Reduce brandy in a saucepan, add white wine, reduce again. Add star anise and tomato paste, roasted shells, and vegetables. Cover with water, simmer 1-1.5 hours. Strain the stock.
3. Prepare sauce: Heat olive oil in a pan, fry garlic, thyme, chili, and spring onions until soft. Add lobster meat, splash of brandy, star anise, tomato sauce, and lobster stock. Simmer 5-8 minutes. Remove thyme and star anise.
4. Cook pasta: Boil tagliatelle in salted water until al dente. Drain, reserve pasta water.
5. Finish dish: Toss pasta with sauce, add pasta water and olive oil for coating. Season with salt, pepper, parsley, and basil.
6. Serve: Plate pasta, top with lobster halves, drizzle sauce. Garnish with basil.