Preheat oven to 500°F. Pat chicken dry. Place chicken, skin side up, in one layer on 17X12-inch rimmed sheet pan or jelly-roll pan with sides. Bake 35 to 40 minutes or until skin is golden brown.
Remove pan liquids to large measure and spoon off any fat that rises to surface. Add enough broth to bring liquid to 4 cups total.
Heat oil in skillet over medium heat. Add onions; cook 3 to 5 minutes or until softened and beginning to brown. Add mushrooms, ginger and garlic; cook and stir 3 to 5 minutes or until garlic is golden.
Add mirin to skillet; boil 1 minute, stirring and scraping up any brown bits from bottom of skillet. Pour into Crockpot™ slow cooker. Stir in miso paste and soy sauce. Add chicken, broth mixture and water. Cover; cook on LOW 8 to 9 hours or on HIGH 4 to 5 hours.
Stir in mustard greens. Cover; cook 5 minutes or until greens are wilted. Serve in shallow bowls with rice, if desired. Garnish with green onions.