Using your hands, gently mix 80/20 Lean Ground Beef (1 pound) with Fine Sea Salt (1 teaspoon), Freshly Ground Black Pepper (1/2 teaspoon), and Ground Allspice (1/4 teaspoon).
Form meat into 4 patties, slightly concave in the center and slightly wider than bun (they will shrink up). Chill patties.
Cut Camembert (4 ounce) into 1⁄4-inch slices, then quarter slices crosswise.
Meanwhile, heat a grill to high (500 degrees to 550 degrees).
Set a medium cast-iron skillet on grill and add Unsalted Butter (2 tablespoon) and Olive Oil (1 tablespoon). When butter melts, add the Onion (1).
Stir, season lightly with Fine Sea Salt (to taste), cover, and cook until onions soften and turn golden, about minutes, stirring every couple of minutes. Scoop onions into a bowl.
Add Unsalted Butter (1 tablespoon) and Olive Oil (1/2 tablespoon) to skillet, then add the Pink Lady Apple (2).
Cook, covered, until golden, about 3 minutes. Stir and cook until tender throughout, about 8 minutes more.
Remove pan from heat, season lightly with Fine Sea Salt (to taste), and set aside.
Reduce heat to medium-high (400 degrees to 450 degrees). Grill the burgers until browned, about 4 minutes.
Turn and cook 2 minutes more, then top each with 2 or 3 slices cheese, cover grill, and let melt 1-2 minutes. Transfer to a baking sheet.
Toast the Hamburger Bun (4), cut side down, about 2 minutes. Transfer to another baking sheet.
Assemble burgers: Slather bottom sides of buns with Mayonnaise (to taste) if you like. Divide apples among bottom halves of buns. Top with burgers, cheese side up, then the onions. Sprinkle with Italian Flat-Leaf Parsley (2 tablespoon) if you like, then add bun tops. Enjoy immediately.