Mix the mayonnaise, Dijon mustard, whole-grain mustard, hot sauce, cornichons, green onion, and parsley in a bowl.
Add salt and pepper to your taste, cover, and refrigerate for at least an hour.
Heat the canola oil in a cast-iron skillet or griddle over medium-high heat.
While it's heating, divide the ground beef into four equal portions and gently form them into patties.
Season both sides of the patties with salt and black pepper.
Place them on the hot skillet and let them cook for 3-4 minutes until they develop a golden-brown crust.
Flip the burgers and add the cheese slices on top.
Cover the skillet to allow the cheese to melt, cooking the patties for another 3-4 minutes to your preferred doneness.
Toast the burger buns lightly.
Spread the prepared sauce on each bun before assembling the burger with lettuce, tomato, or any preferred toppings.