Next time you make lobster, save the shells to prepare this stock. You can also make it with an equivalent quantity of crab shells, or use a combination of crab and lobster shells. For a less expensive option, use shrimp shells. Straining the stock twice makes for a smooth, clean-tasting final product.
Preheat oven to 450°.
Arrange shells in an even layer in a shallow roasting pan. Bake at 450° for 30 minutes or until toasted.
Transfer shells to an 8-quart stockpot. Add celery, fennel, carrot, tomato paste, and onions to pot; stir well to combine. Pour water over vegetable mixture; bring to a boil. Reduce heat, and simmer 4 hours, skimming surface occasionally.
Strain stock through a sieve into a large bowl; discard solids. Strain again through a paper towel-lined sieve. Cool the stock to room temperature. Cover and chill stock. Refrigerate stock in an airtight container for up to 2 days, or freeze for up to 3 months.