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To prep the burgers: In a medium bowl, toss in the ground meat, 1/2 teaspoon salt, 1/2 teaspoon pepper, and Worcestershire sauce.
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Mix together with a fork.
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Once the ingredients have been combined, form four patties.
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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.
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For the lemon garlic aioli: In a small bowl, mix all the ingredients together.
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Place back in refrigerator until burgers are cooked.
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To cook: Preheat grill to around 350.
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If using cast iron grates, use a paper towel or cloth to wipe all grates with canola oil.
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Place burger patties on grill.
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Close lid and prep the buns by brushing canola oil on insides.
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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.
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Add red onions and buns (cut side down).
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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.
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To assemble the burger: Place a handful of arugula leaves on the bottom slice of the bun.
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Place burger with cheese slice on top.
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Add the grilled red onions.
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Slather the aioli liberally on the top half of the bun.