Combine mushrooms and boiling water in a bowl; cover and let stand 20 minutes. Drain mushrooms, reserving 1/3 cup liquid. Rinse and chop mushrooms. Strain the reserved mushroom liquid into a bowl; add chopped mushrooms, broth, and next 3 ingredients, stirring well. Set aside.
Place each chicken breast half between 2 sheets of heavy-duty plastic wrap. Flatten to 1/4-inch thickness, using a meat mallet or rolling pin. Dredge chicken in breadcrumbs.
Heat oil in a large nonstick skillet over medium-high heat. Add chicken; cook 3 minutes on each side or until browned. Add vermouth, and cook 1 minute. Add mushroom mixture; reduce heat to medium, and cook for 10 minutes or until chicken is done.
Note: Dried porcini mushrooms are available in large supermarkets. You can substitute dried shiitake mushrooms; remove stems after soaking.