Line the interior of the Instant Pot Multicooker with parchment paper.
In a large bowl, whisk together the eggs, milk, mustard, salt, and black pepper. Add the bread to the bowl and let sit, tossing occasionally, until nearly all the egg mixture is absorbed, 5 to 8 minutes.
Fold the onion, ham, and cheese into the bread mixture.
Transfer the mixture to the prepared pot, cover with the lid, and press Bake. Adjust the temperature to 350°F and the timer to 2 hours. Cook until the mixture is set, about 1 1/2 hours. Press Cancel.
Use the parchment to transfer the entire egg bake to a serving platter. Serve with a green salad (if using).