Place the English muffin in the toaster and lightly toast. Place the slice of cheese on the English muffin and let it start to melt.
You can encourage it to melt more by sliding it under the broiler for a few seconds. Place the ham on top. Keep warm.
Crack an egg into a small bowl. Bring a small pot of water with the vinegar up to a simmer. Do not bring to a boil.
With a spoon spin the water in the center of the pot to create a whirlpool and very carefully drop the egg into the center of the whirlpool.
It will look like a mess but it will come back together. Simmer the egg, never boil for 3-4 min. depending on the desired doneness.
Go to the 4 min. mark for a sandwich such as shown here or closer to the 3 min. mark if just serving on toast. Salt and pepper and place on top of the ham.