In a blender or food processor, combine the egg yolks, mustard, lemon juice, and mint. With motor running, add the oil in a slow stream and blend until emulsified. Season to taste with salt and pepper and transfer to a bowl.
Spread each slice of bread generously on one side with the mayonnaise. On half the slices, arrange an even layer of cucumber slices. Top with remaining bread slices. Cut each sandwich into quarters before serving.