Add the mayonnaise to a food processor. Add the roasted garlic, discarding the skins. Pulse to combine. With the processor running, slowly drizzle in the tomato-garlic oil until incorporated. Add in the water. Season with salt and pepper to taste.
Heat the grill over high heat. Form the ground beef into 6 patties, each about a ½” thick. Coat each side with a drizzle of olive oil and season well with salt and pepper. Grill the patties for about 2-3 minutes per side, or until slightly charred.
Lower the heat to medium and continue to cook the patties for about 4 more minutes, or until your desired doneness. For the last minute, distribute the gouda cheese slices on top of the patties. Cover the grill and allow the cheese to melt as the meat finishes cooking. Transfer the patties off of the grill and cover with foil.
Meanwhile, drizzle the inside half of each bun with olive oil. Quickly toast the buns on the grill, halved-side down. Toss the arugula with the red wine vinegar, 1 tablespoon of olive oil, and a pinch of salt.
To assemble the burgers, spread a generous amount of garlic mayo onto each bun and distribute the arugula onto the bottom buns. Top with the grilled patties, roasted tomatoes, and fresh basil leaves. Close the burgers with the top buns.
Reserve any extra garlic mayo in an air-tight container in the fridge for up to 7 days.