Prepare Balsamic Infused Portobellos, Peppers and Onions:
Preheat oven to 375ˇF.
Mix first three ingredients together in a glass baking dish. Add mushrooms and toss to coat. Cover with foil and bake for about 25 minutes. Place mushrooms in a baking dish. Let cool and slice.
Heat olive oil in a medium skillet over medium heat. Add onions and saute until very tender and caramelized, add garlic and saute for a minute. Add balsamic and sugar until reduced and absorbed.
In a medium bowl, add mushrooms, onions, chopped red pepper, basil and a splash of olive oil and mix. Add salt and pepper to taste. This can be made a day in advance, covered and refrigerated.
Prepare Balsamic and Rosemary Alioli: Whisk all ingredients together in a small bowl. Reserve
Prepare Burgers: In a large bowl, mix all of the ingredients except the salt and pepper. Form into six patties. (Make them slightly thinner in the middle for even cooking.) Salt and pepper both sides.
Brush the grill with vegetable oil. Place patties on the grill rack and cook for 4 or 5 minutes on each side for medium. During last 2 minutes of cooking, evenly top patties with cheese, tent with foil to help with melting. Place buns, cut side down, on the edges of the grill to lightly toast.
Assemble: Top each toasted bun bottom with a smear of the rosemary alioli, a handful of arugula, top with the burger. Then top with 1/4 Cup of the balsamic mushroom mixture, a drizzle of rosemary alioli, a sprinkle of minced parsley and the toasted bun top.