For a relish for the adult burgers, combine the cherry tomatoes, parsley leaves, minced garlic and olive oil in a small bowl. Season with a pinch of salt, and toss everything together. You can do this ahead of time and chill it in the fridge.
Divide the ground beef into 4 patties. You can make the kid patties slightly smaller, or they can have leftovers.
Dent the center of each burger patty, and divide the white Cheddar between 2 of the indentations and the pimento cheese between the other 2. Doing the best you can, fold the meat up and over the cheese, then lightly press to seal everything together.
Grill or sear the burger patties over medium-high heat, roughly 4 minutes per side. Let the burgers rest for about 5 minutes. Put the burgers in the buns.
Serve the kid (white Cheddar) burgers with apple wedges, chips or fries.
Serve the adult (pimento cheese) burgers topped with the tomato-parsley relish.