1. Heat a large nonstick skillet over medium-low heat. Add 2 teaspoons olive oil to pan; swirl to coat. Add onion and thyme; cook 17 minutes or until golden and very tender, stirring occasionally. Increase the heat to medium-high; add wine, vinegar, and 1/4 teaspoon salt. Cook for 6 minutes or until the liquid almost evaporates, stirring occasionally. Remove onion mixture from pan.
2. Wipe pan clean with paper towels. Heat pan over medium-high heat. Add the remaining 3 teaspoons oil to pan; swirl to coat. Add mushrooms and 1/4 teaspoon salt; sauté for 8 minutes or until mushrooms brown and liquid mostly evaporates.
3. Preheat grill to medium-high heat.
4. Coarsely chop 1 cup onion mixture, and stir chopped onion mixture into beef. Divide the beef mixture into 6 equal portions, gently shaping each into a 1-inch-thick patty. Press a nickel-sized indentation in center of each patty. Sprinkle the patties with remaining 1/4 teaspoon salt. Place patties on a grill rack coated with cooking spray, and grill for 4 minutes on each side or until done.
5. Combine mayonnaise, blue cheese, and garlic in a bowl, and mash well with a fork. Spread the top halves of buns evenly with mayonnaise mixture. Arrange 1/4 cup arugula on bottom half of each bun; top each bottom half of bun with 1 patty, about 2 tablespoons remaining onion mixture, about 1/4 cup mushrooms, and top half of bun.