Let the fillet steaks sit at room temperature for 20 minutes.
Heat the oil in a large skillet and when it is quite hot, add the steaks with salt and pepper to taste and cook over a fairly high heat, turning them every minute, until they are cooked for 6 minutes altogether (for rare), 8 minutes (for medium-rare), and 10 minutes (for well-done).
Meanwhile, put the lemon juice and egg yolks in the container of a blender or food processor and add salt and pepper to taste.
Melt the butter, turn on the machine, remover the insert cap, and drizzle the hot butter in a slow, steady stream.
When it is all incorporated, turn off the machine, add the chopped shallots and tarragon, and turn the machine on for less than 5 seconds just to combine them.
Arrange a fillet steak on each plate and pour the mock Bearnaise on the side of each steak.