In a large bowl, combine all Elk Burger ingredients. Gently mix until just combined – do not overmix.
Form the mixture into 4 patties.
Prepare the Hatch Chile Aioli: In a small bowl, combine all Hatch Chile Aioli ingredients. Mix well and set aside.
Caramelize the Onions: Melt butter in a skillet over medium heat. Add sliced onion and cook, stirring occasionally, for 20-25 minutes, or until deeply golden brown and caramelized. Stir in brown sugar during the last 5 minutes.
Preheat grill to medium-high heat.
Grill the elk burgers for 5-7 minutes per side, or until cooked to desired doneness (internal temperature of 160°F/71°C).
Toast the brioche buns lightly.
Assemble the burgers: Spread Hatch Chile Aioli on both buns. Place arugula and tomato on the bottom bun, top with the elk burger, and add a generous portion of caramelized onions. Finish with the top bun.
