Slice each lobster in half lengthwise, keeping meat intact with shells for moisture retention.
Melt ¼ cup butter in a pan over medium heat; add 2 tbsp lemon juice to the pan for an aromatic base.
Cook lobster halves in the pan until warmed through (3–4 minutes), then discard shells and slice lobster meat into bite-sized pieces.
Add asparagus to the same pan; sauté for 2–3 minutes, turning occasionally. Cover and keep warm.
In a bowl, whisk 3 egg yolks and 1 tbsp lemon juice until pale and frothy.
Place the bowl over simmering water, continuously whisking until thickened.
Gradually incorporate ½ cup melted butter into the egg yolk mixture while whisking constantly.
Stir in ½ tsp smoked paprika, ¼ tsp cayenne pepper, ¼ tsp chili powder, and ¼ tsp salt to create a spicy hollandaise sauce.
Poach 4 eggs: In a saucepan, bring water to a simmer with 2 tbsp white vinegar. Crack eggs into a ramekin, gently slide into water, and cook 3–4 minutes. Remove with a slotted spoon.
Toast English muffins and split them.
Assemble: Place warm asparagus on each muffin half, top with lobster slices and a poached egg. Spread half the Cajun hollandaise over each serving.
Garnish with capers and fresh dill.