Preheat a grill or grill pan to medium-high (400 to 450 degrees F (200 to 230 degrees C)).

Stir together mayonnaise, red onion, ketchup, vinegar, capers, parsley, thyme, and sugar in a small bowl until combined.

Place beef in a large bowl. Add 3/4 teaspoon salt and 1/2 teaspoon pepper. Mix by hand until just combined. Make 12 meatballs by rolling beef mixture between your palms; transfer to a baking sheet. Flatten tops slightly. Drizzle with oil and season both sides with remaining 3/4 teaspoon salt and 1/2 teaspoon pepper.

Arrange patties on oiled grates; grill, uncovered, for 3 minutes. Flip, then top each with 1 piece cheese. Grill, uncovered, until cheese is melted and beef is cooked through, about 2 minutes. Meanwhile, put buns, cut-sides down, on grate; grill until toasted, about 1 minute.

Top bun bottoms with a small spoonful of sauce. Add a patty, shredded lettuce, tomato slices, cornichon slices, and bun tops.