In a blender or using an immersion blender, blend the egg yolk, mustard, and lemon juice until combined.
With the blender running, slowly drizzle in the oil until the mayo is emulsified and thick. Season with salt to taste and keep chilled till ready to use.
Gently fold the crab meat with the homemade mayonnaise, chives, lemon zest, salt, and white pepper. Taste and adjust seasoning.
In a small saucepan, melt the butter over medium heat.
Add the garlic and shallots; sweat until translucent, about 2-3 minutes.
Deglaze the pan with the vermouth, and cook until reduced by half.
Remove from the heat and stir in lemon juice.
Brush the brioche rolls with the warm garlic butter sauce.
Fill them with the crab mixture, drizzle the remaining sauce over the top, and finish with flaky sea salt.