In a medium mixing bowl, mix together all of the ingredients for the burgers without overworking the meat. Divide meat into 3 equal portions and form into ½-inch (12-mm)-thick patties. Use your thumb to make an indentation in the center of each burger. Set aside.
Using a spiral slicer or julienne slicer, cut the parsnips into strings. Preheat a grill to medium-high heat, and preheat a medium saucepan over medium-high heat. Once grill is hot, place the burgers on the grill and close the lid. Cook 6 minutes.
While the burgers are cooking, add the coconut oil to the saucepan. Test the oil by placing one parsnip shoestring in the oil to ensure the oil bubbles. Add the parsnips to the oil and cook for 1 to 2 minutes until golden brown. When the parsnips finish cooking, place them on a paper towel−lined plate to drain.
Flip the burgers and cook for 2 to 3 minutes for medium-rare, or until desired doneness. Remove the burgers from grill and let rest for 5 to 7 minutes.
To assemble the burgers, place each on a bed of lettuce and top with sliced red onions, ¼ cup (60 mL) avocado mousse, and parsnip fries.
