For mustard sauce, in a small saucepan over medium-high heat, bring the 1/4 cup of apple cider to a boil. Cook 4 to 5 minutes or until it's reduced by half. Remove from heat and let cool. Once cider has cooled, transfer to a small bowl; add mustard and tarragon. Stir to combine; set aside
Set up grill for direct cooking over high heat. Oil grate when ready to start cooking.
For burgers, in a large bowl, stir to combine ground sirloin, the remaining 1 tablespoon cider, and the onion soup mix. Form into 4 mini-patties; set aside. Spread cracked pepper on a plate. Carefully press both sides of burgers into cracked pepper.
Place burgers on hot, oiled grill and cook 3 to 4 minutes per side for medium (160 degrees F). To serve, place hot burgers on toasted Kaiser rolls. Top with lettuce, tomato, onion, and mustard sauce.