1. Prepare grill to medium-high heat.
2. Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add mushrooms and next 3 ingredients (through garlic) to pan; sauté 5 minutes or until mushrooms are browned. Add wine; cook 3 minutes or until liquid evaporates. Remove from heat; cool completely.
3. Combine mushroom mixture, salt, 1/2 teaspoon pepper, and beef in a large bowl. Divide mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Place patties on a grill rack coated with cooking spray; grill 4 minutes or until grill marks appear. Carefully turn patties; grill 4 minutes or until desired degree of doneness.
4. Return pan to medium heat; coat with cooking spray. Crack eggs into pan; cook 4 minutes or until whites are set. Remove from heat.
5. Combine remaining 1/4 teaspoon pepper, arugula, cheese, and oil in a medium bowl; toss gently to coat. Combine chives and next 3 ingredients (through mustard) in a small bowl. Spread 2 teaspoons mayonnaise mixture on bottom half of each bun; top with 1 patty. Top each serving with 1 egg, 1/2 cup arugula mixture, and top half of bun. Yield: 4 servings (serving size: 1 burger).