In a shallow bowl stir together flour, marjoram, salt, and black pepper. Place a chicken breast half between 2 pieces of plastic wrap. Using the flat side of a meat mallet, pound chicken lightly to about 1/4 inch thick. Remove plastic wrap. Repeat with remaining chicken breast halves. Lightly coat chicken pieces on both sides with flour mixture; shake off excess.
In a 12-inch skillet cook mushrooms and green onions in 1 tablespoons of the butter over med-high heat until tender; remove 3 from skillet. In the same skillet cook chicken in remaining 2 tbsp butter for 5 to 6 minutes, turning to brown evenly.
Remove skillet from heat. Return mushrooms and green onions to skillet. Carefully add broth and marsala to skillet. Bring mixture to boiling; reduce heat. Simmer, uncovered, for 2 minutes, stirring occasionally. Season sauce to taste with additional salt and pepper. Spoon mushroom on top of chicken when serving on pasta.