Peel and devein ½ # shrimp. Reserve shells. Poach shrimp in a court bouillon of water, half a lemon, 2 bay leaves, and 1 Tablespoon black peppercorns. Remove shrimp just before done, drain, ( do not rinse ) and place on a small sheet pan that has been in your freezer. Return pan to freezer for 10 minutes. Once cool, slice shrimp lengthwise and reserve for garnish.
In a large, heavy-bottomed pot, over medium-high heat, sauté the shrimp and shells until shrimp is just cooked through, and shells have colored. Reduce heat slightly and sweat shallots.
Add brandy and reduce to sec ( dry ).
Add white wine and reduce to sec.
Add tomato paste, and over 3-5 minutes while stirring, cook through.
Add lemon juice, milk and heavy cream and bring to a simmer. Stirring occasionally, add salt, cayenne, and pepper and simmer another 15 minutes. Add tarragon and remove from heat.
Working in batches, transfer bisque to a blender, filling it no more than half full. Cover blender tightly with the lid, and a towel over the cover. Start on the slowest setting, gradually working toward the highest setting. When bisque is completely pureed, strain through a chinois ( fine-mesh strainer ) into a clean pot. Discard solids.
Warm to serving temperature and adjust seasoning and consistency.
Serve and garnish w/ lemon zest, and reserved sliced shrimp and chives.