In a heatproof 24-ounce glass jar that can accommodate an immersion blender, combine the butter, egg yolks, lemon juice, vinegar, 2 tablespoons water, dry mustard, cayenne and ¼ teaspoon salt; cover with the lid and seal tightly. Do not shake or stir.

Place the jar in a straight-sided heatproof container that is about 10 inches in diameter and 5 inches high (a Dutch oven works well). Place a weight on the jar so it will remain in place when water is added to the container, then fill the container with warm water to within about 2 inches of the jar’s lid.

Attach the immersion circulator to the container and set to 165°F. When the water reaches temperature, use a slotted spoon to lower the whole eggs into the water. Cook for 30 minutes. Meanwhile, in a heatproof vacuum-seal bag, arrange the asparagus in a single layer. Seal according to the manufacturer’s instructions.

After the eggs and hollandaise ingredients have cooked for 15 minutes, lower the asparagus bag into the water. If floating, weigh it down with a small plate or bowl. Fill a medium bowl with ice water.

When cooking is complete, use a slotted spoon to transfer the eggs to the ice water; cool for 1 minute. Meanwhile, using tongs, remove the asparagus bag; leave the jar in the water but turn off the circulator. Remove the eggs from the ice bath and carefully peel them; set aside on a clean kitchen towel.

Remove the jar from the water and remove the lid. Using an immersion blender, blend the ingredients, scraping down the sides of the jar as needed with a small silicone spatula, until the hollandaise is smooth and emulsified, about 1 minute.

To serve, place a split English muffin on each serving plate. Place 1 egg on each English muffin half. Carefully open the asparagus bag and, using tongs, divide among the plates. If the hollandaise has separated slightly during standing, blend again until smooth. Spoon sauce over the asparagus and each egg-topped muffin. Sprinkle with tarragon and a pinch of paprika.