Fill a deep skillet or shallow saucepan with water, add a teaspoon of salt.
Set your ChefAlarm probe in place with a pot clip so the tip of the probe is suspended in the water without touching the bottom of the pan.
Set your ChefAlarm high-temp alarm to 180°F (82°C). Once the water comes to a stable target temperature, set the ChefAlarm low-alarm to 175°F (79°C) to make sure it doesn't dip too far during cooking.
One at a time, crack the eggs into a strainer over a bowl or sink, then, once the thin white stops straining out, lower the basket of the strainer into the heated water and tip the egg out.
Jiggle each egg slightly with a wooden spoon for a few seconds to make sure it doesn't stick to the pan.
Set a timer for 4 minutes. You may vary that time depending on your doneness preference and the size of your eggs—3 minutes will be quite runny, 5 will be firmer, like a medium-cooked yolk. When the TimeStick alarm sounds, check the eggs. The yolk should still jiggle slightly under pressure (don't poke it too hard!) and the whites should not be runny.
Transfer them to a bowl filled with warm water, where they can be held easily.
Prepare the ham and muffins
Add 1/2 Tbsp of the unmelted butter to a second skillet and heat it over medium-high heat
Once the butter is hot, add the ham slices as many as can fit at a time.
Cook until browned on one side, flip and brown on the other side. Repeat until all ham is cooked.
Add half the remaining butter to the pan and toast half of the English muffins, face down in the hot buttered skillet.
Repeat with remaining butter and muffins. Alternatively, you can butter each slice and place them on a cooking sheet in a 450°F (232°C) oven for about 5 minutes.