Saute celery, carrots, and onions in olive oil. Add mushrooms, rosemary, and lobster until vegetables are translucent and lobster is pink. Meanwhile, in a separate pot, combine the port and tomato paste and reduce until mixture becomes thick.
Add the milk and cream to the vegetable-lobster mixture and bring to a simmer. Add the reduced port to the cream and whisk to combine. Let simmer and season with salt and pepper.
Make corn starch slurry by whisking equal parts water and corn starch together. A little at a time, add corn starch slurry to soup to thicken mixture a little.
Adjust the seasoning and serve or store for later.
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