Sauté liver in 2 tablespoons butter in a large skillet over medium heat 5 minutes on each side. Remove liver to a warm serving platter.
Cut avocados into 1/4-inchthick lengthwise slices. Sauté slices in 2 tablespoons butter over medium heat 1 minute on each side or until avocado is slightly soft. Arrange slices around liver on platter.
Combine remaining 1/4 cup butter, chicken broth, vermouth, lemon juice, chives, and parsley in a small saucepan. Bring to a boil. Remove sauce from heat; serve over liver and avocado.