Its probably not broken yet.
The limited slip does very little on ice. It uses torque on the spider gears to squeeze the clutches. If your on ice, there is no torque on the spider gears, and no squeezing of the clutch packs.
If you keep roasting one tire though, you will be burning up the differential.
Shovel it out, put some sand (or a rag) under that tire, rock it back and forth, or get someone to push on the top of one of the front tires to help.
2012 V6 Base. 6-speed. 3.31. Roush Wheel Hop Kit. BMR 38mm Front Swaybar. Camber Modded Koni Yellow Struts. Linear Steeda Sport Springs. J&M Camber Plates. J&M Panhard Bar. Airaid Intake. 93R Bama Tune. BBK Ceramic Shorties. Carbon Fiber Trac-Lok w/ Finned Cover. Hawk SR pads. GT Rotors. Spec Al Flywheel & Stage I Clutch, 18x10.5 ET38 Enkei PF01 Wheels. 285 30 R18 RE-71r tires. Boss 302 Radiator.