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1
Peel shrimp, reserving shells.
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2
Coarsely chop shrimp into bite-size pieces, place in a medium container, cover, and refrigerate until ready to use.
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3
Place 1 tablespoon of the butter in a large saucepan over medium heat.
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4
When the foaming subsides, add onions, carrots, celery, and reserved shrimp shells and season with salt and pepper.
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5
Cook, stirring occasionally, until onions are just tender and shells are pink but not browned, about 5 minutes.
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6
Increase heat to medium high and add tomato paste, thyme, and bay leaves; stir until vegetables are coated in tomato paste.
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7
Add water, stir to combine, and bring to a boil.
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8
Reduce heat to low and keep at a lively simmer until broth has a shrimp flavor, about 30 minutes.
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9
Strain through a fine mesh strainer into a heatproof container; set aside and discard solids.
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10
Wipe out any solids in the saucepan and return it to the stove over low heat.
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11
Add remaining 3 tablespoons butter and let melt.
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12
When butter foams, whisk in flour and cook, whisking frequently, until smooth and golden brown, about 4 minutes.
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13
Slowly whisk in Cognac, then reserved hot stock, whisking until smooth, and bring to a boil.
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14
Add cream, cayenne, and reserved shrimp and stir to combine.
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15
Simmer until flavors have melded and shrimp is cooked through but not tough, about 8 to 10 minutes.
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16
Season with salt and freshly ground black pepper, if desired.
