Skewer the 4 onion disks with a toothpick by placing the toothpicks horizontally through each onion disks so they stay intact while on the grill.
In a small bowl whisk together the Holland House Red Cooking Wine, garlic, 1 teaspoon salt, and 1/2 teaspoon pepper.
Pour the mixture to a 1-gallon zipper-lock bag. Add mushrooms and onion disks and seal bag the bag. Gently turn the bag to coat the mushrooms and onions in the marinade, being careful not to pierce the back with the toothpick. Let the veggies marinade for at least 30 minutes or up to 1 hour.
In a small bowl mix together the goat cheese, basil, and a pinch of salt and pepper with a fork. Mix until the basil is evenly distributed (the mixture will be thick like cream cheese).
On a gas grill, turn all of the burners to high and cover the grill until preheated, about 10-15 minutes. Then, reduce the temperature to medium-high and clean and oil the cooking grates. Place the marinated mushrooms on the grill, smooth cap side up, and the onions on the grill. Cover the grill and cook for 5-6 minutes, or until the mushrooms are a little bit charred and they have expelled most of their moisture. Then, flip the mushrooms and onions over, cover, and keep cooking for another 4-5 minutes, or until the mushrooms are charred on the second side.
Using tongs, place the mushrooms and onions to the platter. Remove the toothpicks from the onions.
Spread the goat cheese mixture onto the bottom portion of the toasted Kaiser rolls. Place the portobellos on top of the goat cheese followed by the grilled onions and arugula. Place the Kaiser roll top on the burger and then serve.