Fill a large stockpot, Dutch oven, or sous vide machine with about 2-1/2 inches of water. Turn on the immersion circulator or sous vide mechanism to 172°F. While the water bath is preheating, prepare the jars and the eggs.
Coat all interior surfaces of six 4-ounce mason jars lightly with olive oil or softened butter.
In a large mixing bowl, whisk together the eggs with the half and half, salt, and pepper until frothy. Fold in the shredded cheese and ham.
Ladle or spoon the egg mixture into the mason jars so that each jar is 3/4 full. You want to leave enough room for the eggs to expand or rise. Screw the lids on each jar until secured but not overly tight.
Using tongs, transfer the jars to the preheated water bath. The water should be higher than the jar lids so they are fully submerged. Cook for 1 hour or until the eggs are fully set.
Remove the jars with the tongs and place them on a sheet pan or wire rack to rest until cool enough to handle.
Unscrew the lids. Run a dinner knife around the inside perimeter of each jar to loosen the egg bites. Overturn the jars over a plate and tap to drop the egg bites out. Or, you can eat them right out of the jar. Enjoy while warm.