Heat up your grill or grill pan until it reaches a cozy medium-high temperature of 400 to 450 degrees F (200 to 230 degrees C).
Combine mayonnaise, red onion, ketchup, vinegar, capers, parsley, thyme, and sugar in a small bowl, stirring until mixture is well blended.
In a large bowl, combine beef, 3/4 teaspoon salt, and 1/2 teaspoon pepper by hand. Shape the mixture into 12 meatballs, then place them on a baking sheet. Gently flatten the tops, drizzle with oil, and season both sides with the remaining 3/4 teaspoon salt and 1/2 teaspoon pepper.
Place the patties on the oiled grates and grill for 3 minutes. Flip the patties and add a piece of cheese on top of each one. Continue grilling uncovered until the cheese is melted and the beef is fully cooked, about 2 minutes. While the patties are grilling, place the buns cut-side down on the grate and grill until toasted, approximately 1 minute.
Spread a dollop of sauce on the bottom bun. Layer on the patty, shredded lettuce, tomato slices, cornichon slices, and the top bun.