Finely dice the mushrooms and beetroot by hand, or in a food processor, until roughly chopped. Move to a bowl and add the beef, egg, breadcrumbs, chives and seasoning. Mix well and shape into 4 patties. Leave these to chill in the fridge for 1 hour or the freezer for 15 minutes.
Next, make up the salad. Slice the tomatoes into discs and mix with the red onion. Whisk together the vinegar, mustard and oil with a little pinch of salt and dress the salad. Leave to marinate while you finish the burgers.
Chargrill, or toast, the brioche buns and set aside.
Cook each burger on a hot cast-iron griddle pan for about 6 minutes on each side (longer or shorter depending on how you like them cooked) then carefully sear the edges, rolling the patties on their sides gently in the pan.
Serve the patties in the toasted buns with the tomato and red onion salad.