Place chimichurri ingredients in a food processor and fine chop into a thick sauce.
Place meat in a bowl and season with salt and pepper. Add the chimichurri and combine. Form 6 patties and chill them a few hours for the flavors to combine.
For the toppings, heat a skillet over medium-high heat with oil. Add onions and sweat a minute or two; add cabbage, season with salt and pepper and let wilt 2-3 minutes more. Remove from heat.
Heat a griddle pan or cast-iron skillet over medium-high heat with a drizzle of oil. Add tomatoes, cook a minute on each side then remove to plate. Brush rolls with melted butter and griddle them until toasty and golden, remove to a plate. Add another drizzle of oil and cook patties 8-10 minutes.
If making your own aioli, whisk up the egg yolk, Dijon, lemon juice then stream in grapeseed oil slowly down side of bowl while whisking continuously, season with sea salt to taste. Stir in ketchup, Worcestershire, relish and combine.
Place burgers on rolls and top with onions and cabbage, tomato and some sauce.