In a bowl, mix together all ingredients for the lamb burger. Form into patties.
Heat a pan over medium heat (can use a grill pan). Add olive oil to cover the bottom of the pan. If using grill pan, use oil spray. Add burgers, cooking for 3-4 minutes on each side. Remove and let rest for 2-3 minutes before enjoying.
To make the aioli, combine all aioli ingredients in a bowl.
Toast brioche bun.
Assemble burger: bun, red onion, tarragon aioli, arugula, burger patty, sunny side up egg, and the top bun!