Char leeks over gas stove flame or grill until the outer layer is somewhat scorched but not burnt to a crisp. Remove from the flame and cut finely.
Melt 1 tbsp of butter in a small sauce pan over medium heat. Add the charred leeks along with a pinch of salt and pepper. Cook for one minute, then add vinegar. Cook until vinegar has evaporated, then set aside.
Preheat a grill to medium-high heat. Brush grill racks with canola oil and add steaks. Cook 6 minutes per side for medium rare. Remove steaks from grill and let rest.
Melt the full stick of butter. In a blender, puree the egg yolks, lemon juice, and a pinch of water until fully incorporated. Turn the blender to the lowest setting and slowly add the melted butter until a creamy sauce develops. Pour sauce into a bowl and add in leeks and tarragon.
Plate the steaks and drizzle with the leek béarnaise sauce. Add salt and pepper to taste.