Place the chimichurri ingredients in a food processor and fine chop into a thick sauce.
Place the meat in a bowl and season with salt and pepper. Add the chimichurri and combine. Form six patties and chill them for a few hours for the flavors to combine.
For the toppings, heat a skillet over medium-high heat with the oil. Add the onion and sweat a minute or two; add the cabbage, season with salt and pepper and let wilt for 2-3 minutes more. Remove from the heat.
Heat a griddle pan or cast iron skillet over medium-high heat with a drizzle of oil. Add the tomatoes; cook for a minute on each side, then remove to a plate. Brush the rolls with melted butter and griddle them until toasty and golden; remove to a plate. Add another drizzle of oil and cook the patties for 8-10 minutes.
If making your own aioli, whisk up the egg yolk, Dijon and lemon juice, then stream in the grapeseed oil slowly down the side of the bowl while whisking continuously; season with sea salt, to taste. Stir in the ketchup, Worcestershire, relish and combine.
Place the burgers on the rolls and top with onions and cabbage, tomato and some sauce.