Combine yogurt, mustard, and salt in a small saucepan; whisk gently on low heat until warm and smooth. Remove from heat.
Fill a large skillet with 2 inches of water and 1 tablespoon vinegar; bring to a boil and reduce heat to a simmer. Crack one egg into a small dish and gently pour it into the skillet, doing the same for each additional egg, one by one. Cook for 3-5 minutes until desired doneness is reached, spooning hot water over the yolk until slightly opaque. Carefully remove eggs from the water with a slotted spoon, letting the excess water drain away.
Toast English muffins (or have the kids do this part!), and place a toasted muffin half on each plate with a piece of ham on top.
Place an egg on top of the ham and drizzle yogurt sauce over the top.
Heat oil in a skillet over medium heat. Trim off the bottom two inches of the asparagus; discard ends. Add asparagus to the skillet and cook until slightly tender. Serve on the side.