Heat up a griddle pan to a medium high heat.
Add the four halves of the brioche bun onto the griddle pan, with the centre facing downwards. Griddle for 2-3 minutes, or until griddle marks start to form. Set aside and place on a wire rack to stop the buns going soggy.
Slightly turn down the heat, and carefully pour in the olive oil. Add the slices of red and white onion, season with a little sea salt and freshly milled black pepper, and sweat down the onions until soft. If space (and if not cook in a separate batch) add the sliced Portobello mushrooms. Season lightly and griddle for 3-5 minutes, until they have reduced in size and softened.
Transfer onions and mushrooms to a plate and set aside. Increase the heat of the griddle pan to a high setting (8/9 on my induction hob).
Season lightly the minced beef patties and add to the griddle pan. Griddle for 4 mins and occasionally press down gently with a suitable utensil, then carefully turn over. Griddle the other side for a further 4 minutes or so. Flip the burger over one more time, then place the slice of Fol Epi cheese until it starts to melt.
Take the bottom of the toasted brioche bun, and add the red and white onions.
Next, place the burger patty on top of the onions. Layer on top the Portobello mushrooms, vine-ripened tomato, and finish with the slices of cornichons.
Finally, spread a thin layer of the Dijon mustard on the underside of the top of the brioche, and place on top.
Enjoy.