Cast iron, beef, and egg.
Cutting the hole
Use a fairly large patty, since the egg will take up quite a bit of space. You want it to be at least 1/2 an inch thick.
Press and twist
Push a cup or a ring cutter into the center of the patty and twist it around until the center separates. Carefully pick up the doughnut-shaped patty, but don't get rid of the smaller patty in the center—you'll use it later.
After seasoning with salt and pepper, lay the doughnut patty in a ripping hot cast iron skillet with a bit of vegetable oil (or beef fat). Let it cook for about a minute undisturbed, then...
Add the egg
If your egg is too big, it may overflow a bit. That's ok, this is not a not-messy operation.
Flip and continue
Carefully flip the patty (the egg has a tendency to leak out, so do it fast), and continue to cook on the second side. You're aiming for a liquid yolk and medium rare patty here.
The cheese covers up the egg, disguising the deed.
Assemble the burger with whatever toppings you'd like (though sauces probably aren't necessary—this burger provides its own).
This is not recommended. Much better is to...
...serve it to an unsuspecting guest.
Tasty, tasty surprise.
What about the minis?
And remember those two tiny patties you cut out from the center of the larger ones?
Get your buns
Thanks to the folks at the White Castle down the street, we can put those tiny patties to use.
In case you weren't full enough
Slide on by.