Combine the broth, milk, corn grits and melted butter in a medium mixing bowl. Cover and allow the grits to soak for at least two hours, up to eight hours.

Stir in the shredded cheddar cheese, salt and pepper to the grits. Spoon the soaked grits into the Mason jars. Do not fill the jars more than 2/3 full.

Add one cup of water to the bottom of the Instant Pot. Set a trivet inside. Arrange the filled jars on the trivet.

Seal the lid on the Instant Pot. Select the MANUAL setting and set the time for 20 minutes. Afterwards, let the pressure release naturally for five minutes, then manually release the rest of the pressure.

Check the water level in the bottom of the Instant Pot and add more water as needed to maintain at least one cup.

Carefully crack one egg into each jar. Replace the lid of the Instant Pot and seal. Select MANUAL again and set the time to 0 for egg whites that are just set and a runny yolk. For more well-done eggs, add an additional minute to the cooking time. More than three minutes will result in hard-boiled eggs.

As soon as the time is up, immediately release the pressure manually.

Garnish with salt, pepper and diced green onions or chives. Serve immediately.