1. Combine broth and 1 cup water in a saucepan over medium heat; bring to a simmer. Heat oil in a large skillet over medium heat. Add mushrooms; cook 5 minutes or until moisture evaporates. Add shallots; cook 4 minutes, stirring frequently. Add fregula and salt; cook 1 minute, stirring constantly. Stir in wine; cook 30 seconds or until liquid is nearly absorbed, stirring constantly.
2. Set aside 1/4 cup broth mixture; cover and keep warm. Add remaining broth mixture, 1/2 cup at a time, stirring constantly until each portion of broth is absorbed before adding the next (about 15 minutes total). Remove from heat.
3. Stir in reserved 1/4 cup broth mixture, chives, goat cheese, and 1 1/2 teaspoons abbamele. Sprinkle with walnuts. Drizzle each serving with 1/4 teaspoon abbamele. Serve immediately.