Preparation

Poach the lobster

• Melt the butter in a medium saucepan over low heat until it begins to simmer, 7 to 8 minutes. Add the garlic, lemon juice, and 1/2 tsp. salt. Return to a simmer over low heat, stirring occasionally.
• Meanwhile, remove the lobster meat from the shells. Cut the lobster into bite-size pieces, and add it to the simmering butter. Poach the lobster, maintaining a gentle simmer over low heat and stirring occasionally, until the meat is opaque and cooked through, 5 to 6 minutes. Remove the pan from the heat, cover, and set aside to keep warm.

Make the sauce

• In a small bowl, beat the yolks and lemon juice with a fork. Transfer the mixture to a blender, and blend on high speed until frothy and bright yellow, about 1-1/2 minutes. With the blender on high speed, slowly drizzle in the warm poaching liquid, and blend until the mixture thickens, about 1-1/2 minutes. Transfer to a medium bowl, and fold in the cayenne and tarragon. Season to taste with salt. Set aside.

Poach the eggs and serve

• Fill a medium saucepan two-thirds full with water, stir in the vinegar, bring to a boil over high heat, and then reduce to a gentle simmer.
• Meanwhile, put one English muffin half cut side up on each of 4 serving plates. Drizzle about 2 tsp. of the remaining warm poaching liquid over each muffin.
• Break the eggs into separate shallow cups or ramekins, and slide into the water. Poach the eggs, maintaining a gentle simmer, until the whites are firm but the yolks are still runny, 3 to 4 minutes. Remove the eggs from the water with a slotted spoon, gently blot the bottoms dry with paper towels, and transfer to the muffins. Divide the lobster evenly among the plates, spooning it over the eggs, and top the lobster with about 2 Tbs. of the sauce. Serve immediately, passing the remaining sauce at the table.