Cook the ham in a small skillet over medium heat until browned on both sides. Set aside.
Fill a medium saucepan with a tight-fitting lid with 2 inches of water. Add the vinegar and bring to a boil. Meanwhile, crack each egg into separate small heatproof mugs. Bring the boiling vinegar water to a simmer and gently lower the mugs all the way into the water. Carefully remove the mugs and leave the eggs behind. Cover and simmer the eggs for 3 minutes. If the yolks appear underdone for your liking, continue cooking them for 1 minute more. Remove the eggs with a slotted spoon to a paper-towel-lined plate.
While the eggs simmer, combine the wine and lemon juice in a saute pan and bring to a boil over high heat. Let boil 1 1/2 minutes. Reduce the heat to low, then whisk in the butter, piece by piece, allowing each to emulsify before adding the next. Turn off the heat and season with salt and pepper.
Top each English muffin half with a piece of ham, followed by an egg. Drizzle with the sauce and sprinkle chives over the top. Serve warm.
Adapted from "One Dish at a Time: Delicious Recipes and Stories from My Italian-American Childhood and Beyond" by Valerie Bertinelli © Rodale 2012. Provided courtesy of Valerie Bertinelli. All rights reserved.