Prepare the Chive Biscuits

1. Whisk together flour, baking powder, sugar, and salt in a large bowl to combine. Add butter; using your fingertips, smear each cube of butter into the dry ingredients. Toss to evenly distribute butter into dry ingredients. Add cream, buttermilk, and chives. Using your hands, combine wet ingredients into dry ingredients by tossing very lightly and gently, adding more buttermilk if dough is too dry.

2. Transfer dough to a lightly floured work surface and pat out to a 7- x 5-inch rectangle. Using a bench knife, fold one-third of dough lengthwise over onto center of dough. Fold the remaining third of dough over on top, folding the dough up like a letter. Roll out dough to 1/2-inch thickness. Using a 3-inch round biscuit cutter, cut out six biscuits. Place biscuits on a rimmed baking sheet. Cover and freeze until firm, about 15 minutes. Meanwhile, preheat oven to 375°F.

3. Brush frozen biscuits evenly with cream and sprinkle with additional salt, about 1/4 teaspoon or more to taste. Bake in preheated oven until biscuits are puffed and barely golden brown, 20 to 24 minutes. Remove from oven and set aside.

Prepare the Lobster Gravy

1. Melt butter in a small skillet over medium. Add flour; cook, stirring often, until pale golden in color, about 2 minutes. Transfer to a small bowl and set roux aside.

2. Working in two batches, cook lobsters in a large pot of boiling salted water until tail meat is firm and opaque, 12 to 18 minutes per batch. Remove from water. Let stand until cool enough to handle, about 10 minutes. Using a mallet and kitchen shears, crack lobster shells and remove meat from tails and claws. Coarsely chop lobster meat and transfer to a large bowl; cover and refrigerate until ready to use. Reserve all shells; discard gills.

3. Heat oil in a separate large pot over medium-high. Add lobster shells; cook, stirring often, until well browned, about 6 minutes. Add fennel, onion, carrot, and celery; cook, stirring often, until lightly caramelized, about 15 minutes. Add tomato paste; cook, stirring often, until color becomes a reddish brown, about 2 minutes. Add wine, stirring to scrape up any browned bits from bottom of pot. Cook, undisturbed, until wine is reduced by half, about 5 minutes. Add broth, paprika, cayenne, star anise, and roux. Whisk to incorporate the roux. Bring to a simmer over medium-high. Reduce heat to medium-low; simmer, stirring occasionally, for 1 hour.

4. Pour lobster sauce through a strainer into a large heatproof bowl and discard solids; return sauce to pot. Bring to a boil over medium-high; boil, stirring often, until reduced to sauce consistency and coats the back of a spoon, about 10 minutes. Whisk cold butter into sauce. Add reserved lobster meat and bring to a simmer over medium. Simmer, undisturbed, until heated through, about 5 minutes. Stir in salt.

Prepare the Turmeric-Poached Eggs

1. Bring water to a simmer in a medium pot over medium. Whisk in vinegar and turmeric. Crack each egg into a small individual bowl. Working with one egg at a time, gently slip eggs into simmering water. Cook eggs until whites are set but yolks are still runny, 3 to 4 minutes.

2. Using a slotted spoon, transfer eggs to a paper towel-lined plate to soak up any water. Eggs may be covered with plastic wrap and kept at room temperature up to 1 hour, or covered and stored in refrigerator up to 1 day (reheat in simmering water until warm, about 1 minute).

Assemble the plate

1. Cut Chive Biscuits in half horizontally to split. Arrange biscuits evenly, cut sides up, among six plates. Top evenly with Lobster Gravy and Turmeric-Poached Eggs. Garnish with chives and black pepper.