Steaks: Heat a 10-inch cast-iron skillet over medium-high until it is very hot but not smoking, about 2 minutes. Pat steaks dry with paper towel. Season steaks on both sides with salt and pepper. Add half the butter to the skillet and set one of the steaks directly on top. Repeat with remaining butter and steak. Without moving the steaks, sear until steaks release easily from pan and a golden brown crust has formed, about 4 minutes. Using tongs, turn steaks over and continue cooking until an instant-read thermometer reaches 115 to 120 degrees for rare, 125 degrees for medium-rare, and 135 to 140 degrees for medium. Transfer steaks to a warm plate to rest.

Sauce: Remove pan from heat and carefully pour in vermouth (it will spatter). Return skillet to heat. Deglaze pan, stirring up any browned bits with a wooden spoon, and cook until liquid is almost completely reduced, about 45 seconds. Add cream and any juices that have collected from steak; stir to combine. Stir in mustard; heat 15 seconds. Cook until sauce lightly coats the back of a spoon, about 10 seconds. Season with salt and pepper.

Place steaks on dinner plates and pour sauce over the top, dividing evenly, before serving.