To prep the burgers: In a medium bowl, toss in the ground meat, 1/2 teaspoon salt, 1/2 teaspoon pepper, and Worcestershire sauce.
Mix together with a fork.
Once the ingredients have been combined, form four patties.
To prep the red onion: Brush extra virgin olive oil on both sides of all the slices and then sprinkle with some salt and pepper.
For the lemon garlic aioli: In a small bowl, mix all the ingredients together.
Place back in refrigerator until burgers are cooked.
To cook: Preheat grill to around 350.
If using cast iron grates, use a paper towel or cloth to wipe all grates with canola oil.
Place burger patties on grill.
Close lid and prep the buns by brushing canola oil on insides.
After a few minutes, check the patties. If the bottom half looks just about cooked, roughly 3 minutes, flip over, and place cheese on top.
Add red onions and buns (cut side down).
Close lid. After another 3 or so minutes, remove everything from grill. The cheese won’t be completely melted, because of the thickness, but it’s okay, that’s the point ;-) The cheese will stay extra sharp if the inside isn’t melted. On the red onions, you’re looking for good char marks on one side, but the onion still a bit crunchy. For the buns, you’re looking to get the bottoms nice and charred, but not so much as to make the bread tough.
To assemble the burger: Place a handful of arugula leaves on the bottom slice of the bun.
Place burger with cheese slice on top.
Add the grilled red onions.
Slather the aioli liberally on the top half of the bun.