Stop me if this sounds familiar: your son or daughter loves cheese—but also can be bafflingly finicky about how he/she eats it. Maybe he/she loves string cheese but hates stringy cheese; maybe he/she loves tacos but hates quesadillas, or vice versa. There’s probably no logic—or any consistency—to this antipathy, which you no doubt learned from serving your child something he/she should so obviously like, only to watch your kiddo reject it for utterly nonsensical reasons.
But you will want to serve your child a cheeseburger at some point, and there is a way to do it that can lead to elation rather than rejection: the Juicy Lucy. Essentially a stuffed burger, with the cheese inside the patty rather than on top, think of it as the ultimate form of “hiding” foods: with the cheese melting inside the meat, your child gets all the visceral pleasure of melty cheese with none of the randomly squeamish visuals. Restaurant versions of the dish often yield a molten center, the cheese often squirting—if not exploding—out upon biting into the center; for obvious reasons, we have tempered that, creating more like a strata of not-too-hot cheese lava in between the meat. It looks cool; it takes only a few minutes more of prep to pull off than our standard burger; and you get to eat your cheeseburger, too. Only wins here.
Serve based on how your child is likely to react to melted cheese visible at the center of the burger. Cutting it in half helps cool the center quicker but may repel the cheese phobic; cutting into quarters or smaller pieces will give your child a mix of the usual and novel pieces, which may or may not be a good thing. Definitely provide the usual condiments they usually eat with the burger.
While we used cheddar and colby jack cheeses in the burger shown in the recipe photos, if you really want some oozy, almost liquid-y cheese to come out the center, use American or even Velveeta cheese, both of which are extremely, well, melty.