Heat the olive oil in a small frying pan over a medium heat. Add the shallots and cook for about five minutes, or until just softened. Remove the shallots from the pan and set aside to cool.
In a large bowl, using your hands or a wooden spoon, mix the shallots with the crabmeat, parsley, lemon zest, eggs, mayo (or Vegan mayonnaise substitute), salt, freshly ground black pepper and breadcrumbs until well combined.
Shape into four equal-sized patties. Wrap each burger tightly with clingfilm and leave to chill in the fridge for at least two hours, or overnight.
Heat the safflower oil over a medium-high heat in a large frying pan. Cook the burgers for 1 minute on each side, until golden-brown.
For the spicy remoulade, combine all the remoulade ingredients in a small bowl.
Squeeze some of the lemon juice over each burger and serve on the toasted brioche buns spread generously with spicy remoulade.