Preheat grill or grill pan over high heat. In a bowl, season ground beef with 1/8 teaspoon each salt and pepper. Combine with hands; roll into 2-tablespoon patties, lightly flattening the tops. Place patties on a plate; drizzle with olive oil, and season each side with 1/8 teaspoon salt and pepper. Place burgers on hot grill. Cook 3 minutes; turn over with tongs. Top each burger with 1 slice Cheddar; allow cheese to melt. Separate each mini brioche bun. Place on grill; toast about 1 minute per side. Slice cherry tomatoes into 3 slices each and cornichons into 4 slices each. Sandwich each toasted bun with 1 burger (cheese side up), 1 arugula leaf, 1 tomato slice, and 1 cornichon slice.