Special equipment: a deep-frying thermometer

Whisk together the ketchup, mayonnaise, mustard and pickle in a small bowl (this is the special sauce).

Cut a 3/4-inch-thick slice off the ends of each onion. Peel and slice the onions into 1/2-inch-thick rounds. Separate the rings and set aside the 8 largest rings. Set aside another 8 rings that are about 1 inch smaller in diameter. (Reserve the remaining onion for another use.) Mix together the beef, cheese, 1/4 cup special sauce, 1 1/2 teaspoons salt and 1/4 teaspoon pepper in a medium bowl until combined.

Put the large onion rings on a parchment-lined baking sheet. Pack a teaspoon of the meat mixture evenly around the inside of each ring, leaving a hole in the middle. Nest a smaller ring in the middle of each to create the shape of a doughnut. Freeze until completely hard, about 1 hour. 

Whisk together the milk, oil, baking powder, onion powder, egg, 1 cup flour and 1/2 teaspoon salt in a small bowl until smooth. Put the remaining 1/2 cup flour in another small bowl.

Fill a large pot with 3 inches of oil and attach a deep-frying thermometer to the side. Heat the oil to 375 degrees F. Working in batches, dredge each stuffed ring into the flour, shaking off the excess. Then dip into the batter, letting any excess drip back into the bowl. Sprinkle with sesame seeds and carefully slip into the hot oil. Fry, turning once, until the crust is golden brown, puffed and crisp, 2 to 3 minutes per side. Return the oil to 375 degrees F after each batch.

Transfer the cheeseburger onion rings to a paper towel-lined baking sheet. Serve with the remaining special sauce for dipping.