Preheat the barbecue to medium-high heat and oil the grill.
In a bowl, mix the kale, onion, oil, lemon juice, and basil. Season with salt and pepper. Set aside.
In another small bowl, mix the Veganaise, garlic, and chives. Set aside and refrigerate.
Lower the barbecue to medium heat and grill the Vegeat burgers on each side until they are cooked through to an internal temperature of 74 °C (165 °F). Season with salt and pepper. Grill the burger buns.
Top the burger with the vegan aioli and marinated green kale. Serve with a watermelon salad.