Directions for Aioli
1. In a small bowl whisk together the garlic, egg yolk, lemon juice and water. Slowly begin to add oil to emulsify until all the oil is used or the desired thickness is reached. Season with salt.
Directions for Assembling
In a small pot add vinegar, shallot, black pepper, tarragon sprigs and reduce to 2 tbsp. strain through a mesh sieve. Place the vinegar in a steel bowl over a water bath. Add 1 tbsp water. Add two yolks. Whisk over the water bath until the egg mixture is thick and ribbon like. Remove from heat and slowly whisk in melted butter until thick. Add chopped tarragon and season to taste (about a ½ tsp)
Season beef generously and then sear in a pan or grill burger to desired temperature.
Butter the inside of the brioche bun and griddle until golden brown.
Once you flip the burger slather the top of the burger with onion compote about 1.2 tbsp. Then cover with gruyere cheese. This might melt by the time the burger is done. If not broil for a couple seconds to finish.
Begin to cook four eggs sunny up in egg pans if possible to hold the round shape.
Smear the aioli over the buttered buns to cover.
Equally distribute the arugula over the 4 bottoms.
Place the burgers on the arugula and then top with a fried egg on each.
Last spoon béarnaise over the top of the burgers and serve with a steak knife and fork!