In a pot on the stovetop, bring the brine ingredients to a quick boil. Remove from the heat and allow to cool. Once cooled, add the rinsed heart and brine to a container, fully submerging the heart in the brine. Cover and refrigerate up to 24 hours.
Get a skillet hot on the stovetop. Add the oil, butter, smashed garlic, and fresh thyme. Add the filets and press with the weight of a bacon press or smaller skillet. Cook for just two minutes.
Flip the filets and tilt the skillet, gathering all of the liquid into one corner. Use a spoon to baste the filets with the hot oil butter mixture for 1 to 2 minutes. Remove from the skillet and set on a sheet of tinfoil and close loosely, allowing it to rest for 5 to 10 minutes. Slice against the grain and serve.
Venison heart is very tender but dense. I liken it to a cross between a high-end beef steak and a pork loin. There is little to virtually no fat requiring the cooking method to either be low and slow as a full heart that has been trimmed, or with a quick sear like a typical steak. Aim for a rare to medium-rare treatment of the filets.