Heat olive oil in a large nonstick skillet over medium heat. Add leek, white onion, and green onions; cook 5 minutes, stirring frequently. Cover, reduce heat to low, and cook 20 minutes or until leek and onions are golden brown, stirring frequently.
While onion mixture cooks, prepare pasta according to package directions, omitting salt and fat.
Bring the broth to a boil in a small saucepan. Add dried currants and vinegar. Add broth mixture, cream, and salt to onion mixture; cook 2 minutes, stirring frequently. Remove from heat; stir in arugula, mozzarella, and thyme. Add to pasta; toss gently to combine.