People are touchy about how their meat is handled. That didn’t sound right. Anyway, here’s what I would do as a non-grill-owner.
Divide each of the 1/2# patties into 1/4# patties and very gently shape them to the thickness you want.
Arrange them about an inch apart in a baking dish (you can line it with aluminum foil for easy clean up) that’s at least 9x9" in dimension and make a divot in the center of each patty that almost goes down to the bottom of the dish. (It’ll cook more evenly and expand to fill this hole by the time it’s done. If you like things rare, skip this part.)
Stick the dish in the oven on the middle rack and cook about 15 minutes at 350 F, then turn them all over and cook the rest of the way to the done-ness you prefer (depending on how thick you make them, it could be 5, 10, 15, or 20 minutes). You can cut one patty open to check inside periodically (sacrificial patty). If you want melted cheese on top, add it during the very last couple of minutes of cooking.
When they’re done, scoop them all into a heat proof container right away and then pour the drippings over the top or save it in another heat-proof container. You can freeze them for later but I have a feeling they’ll be gone in a few days–Om, nom, nom!
(Make all my sauteed toppings suggestions above by chopping things so that they’re no more than 1/2" thick, adding 1 - 2 tablespoons of the fat to each cup of the chopped stuff, and stirring slowly on low-med heat in a saucepan until things look and taste the way you want. You’ll know things are carmelized when they turn a light brown and start to taste and smell sweet.)