Melt butter in a large stockpot over mediumhigh heat. Saute shallots until golden, about 3 minutes. Add mushrooms and cook over high heat until slightly browned, about 2 additional minutes.

Remove from heat and add brandy and Madeira, reserving 1/4 cup of each for final seasoning. Cook over high heat until liquid is reduced by half. Add lobster stock and return to a boil. Reduce to a simmer and cook, uncovered, 15 to 30 minutes, depending on the strength of your stock. (Weaker stocks need to cook longer to concentrate their flavors.) Occasionally skim and discard foam that rises to the top.

Make beurre manie by mixing butter and flour with your fingers to form a smooth paste. Press onto ends of a whisk.

When broth has a strong, almost salty, lobster flavor, add cream and bring it back to a boil, skimming foam occasionally. Then whisk in beurre manie until dissolved. Cook an additional 5 minutes over high heat, whisking occasionally.

Strain through a fine sieve, pressing with the back of a ladle to extract all juices. Season with lemon juice, reserved brandy and Madeira, and Tabasco pepper sauce, and serve immediately.

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