Rub the eye of round roast with seasonings and sit at room temperature for 40 minutes. Heat grill to medium temperature. Cook the meat uncovered on low rack for 12 minutes, turn grill 12 more minutes, turn grill 15 more minutes. Let set for 15 minutes. Then slice thin. Fix Bernaise according to package. Blend avocado in a blender, add sauce gradually. Serve hot with meat.