Squeeze ½ tsp (2.5 ml) lemon juice into a medium-size bowl. Whisk in next 7 ingredients. Add more lemon juice, if needed.
If making ahead, make to this point, then cover and refrigerate for up to 2 days. Bring sauce to room temperature before using.
Lightly oil Hen & Chick Egg Poachers (set of 2). Place Poacher in Fry Pan with about 2” (5 cm) water and bring to a boil. Reduce to a simmer, crack eggs into the Poacher, and float in water. Cover pan with lid and simmer for 3–5 minutes, until desired firmness.
Meanwhile, heat bacon in microwave until warm.
To serve, place English muffins (open) on plates and top with bacon slices. Using a spoon, remove Poacher from water. Run spoon around the egg to unmold, and place egg on bacon slices. Spoon sauce over eggs and sprinkle with chives.