For the burgers: In a shallow baking dish, use a whisk to combine the lemon juice, olive oil, Worcestershire, thyme, and garlic powder. Place mushrooms in the marinade, cover, and refrigerate for at least 1 hour. Turn over the mushrooms, cover again, and refrigerate at least another hour. (Mushrooms can marinate for several hours.)
For the onions: In the meantime, make the onions. In a skillet, heat the olive oil over medium heat. Add the onions and cook about 5 minutes, stirring occasionally. Add the water, cover, and reduce heat to low. Cook 20 to 30 minutes, stirring occasionally, until the onions are golden brown. Remove the pan from the heat. Season the onions with salt and pepper. Set aside.
For the aioli: Combine all ingredients in a small bowl. Refrigerate until serving time.
To finish the burgers: Preheat oven to 450 degrees F. Remove mushrooms from marinade and season with salt and pepper on both sides. Place the mushrooms on a baking sheet, gill sides up. Bake for 5 minutes. Remove from the oven and arrange cheese and caramelized onions on the gills of each mushroom and bake another 5 minutes, until the cheese is melted and bubbly. Place 1 mushroom on each muffin and garnish with aioli and arugula. Serve immediately.
Courtesy of Food Network Magazine