How to Make It
Stir together mayonnaise, shallot, cornichons, chives, green peppercorns, cognac, and lemon juice in a medium bowl; season to taste with salt. Chill.
Cut chuck and rib eye into 3- x 3/4-inch strips. Follow instructions for grinding beef using the grinding technique here through step 5.
Shape ground beef into 6 tightly packed (4-inch-wide) patties. Heat 2 teaspoons oil in a large skillet over medium-high; add 3 patties, and season tops evenly with 1/2 teaspoon salt. Cook until well browned on bottoms, about 4 minutes. Flip burgers, and season evenly with 1/2 teaspoon salt and 1/2 teaspoon black pepper; cook to desired degree of doneness, about 4 minutes for medium. Transfer patties to a plate, and tent with aluminum foil to keep warm. Wipe any excess salt from skillet. Repeat with remaining 2 teaspoons oil, remaining 3 patties, remaining 1 teaspoon salt, and remaining 1/2 teaspoon pepper.
Serve patties on brioche buns with sauce, red onion, and crispy fried onions.