Prepare the aioli by whisking together mayonnaise, lemon juice, lemon peel, mustard, and vinegar in a small bowl.

Gradually whisk in olive oil into the aioli.

Season the aioli to taste with salt and pepper. Chill until ready to serve.

Whisk egg yolks and creme fraiche in a small bowl.

Combine crabmeat, chives, shallot, and chopped parsley in a medium bowl.

Gently mix in mushrooms and yolk mixture.

Beat egg whites in a medium bowl until stiff but not dry.

Fold the beaten egg whites into the crab mixture.

Melt butter in a heavy skillet over medium heat.

Drop 1/4 cup of crab mixture into the skillet for each fritter, spacing them apart.

Sprinkle each fritter with salt and pepper, and flatten slightly.

Cook the fritters until the bottoms are brown, about 3 minutes.

Turn the fritters over and cook until cooked through, about 3 minutes longer.

Transfer the fritters to plates.

Top each fritter with a dollop of aioli.

Prepare the salad by whisking oil and lemon juice in a medium bowl.

Season the salad dressing with salt and pepper.

Mix in fresh basil and parsley leaves.

Place the herb salad alongside the fritters and serve.