Prepare the Lobster: Bring a large pot of salted water to a boil. Add the lobster tails and cook for about 6–8 minutes until shells are bright red. Chop lobster meat into bite-sized chunks once cooled.

Toast the Brioche: Slice brioche buns in half and heat a skillet with butter. Toast each half for 2–3 minutes per side until golden brown. Keep warm.

Poach the Eggs: In a saucepan of gently simmering water, add vinegar. Crack eggs into small bowls and slide into water. Poach for 3–4 minutes until whites are set. Drain excess water.

Make the Hollandaise: Whisk egg yolks and lemon juice in a heatproof bowl over simmering water. Whisk for 3–5 minutes until thickened. Gradually add melted butter and Cajun seasoning.

Assemble the Dish: Place a toasted brioche half on each plate, top with lobster meat and poached egg, then drizzle with hollandaise.

Garnish and Serve: Sprinkle with chives and cayenne pepper if desired. Serve immediately.