On a cutting board, cut the rabbit into 9 pieces with game shears. Cut the saddle (center portion) into 3 pieces. Cut the front portion (front legs) in half through the backbone. Chop each hind leg into 2 pieces.
In a 2-quart pot or larger, add Tender Quick, brown sugar, 2 tablespoons of the pickling spice, and garlic to the and boil for a few minutes until tender quick and brown sugar are dissolved.
Remove from heat and let cool completely in the fridge. Once it has cooled, place the rabbit pieces in a 1-gallon Ziploc bag and add the brine. Seal and lay flat inside a 9 x 13 casserole dish and place in the refrigerator for 4 days. Check daily to make sure the rabbit pieces are completely submerged and stir the brine.
After 4 days, remove the rabbit pieces from the brine and rinse well under cool water.
Transfer rabbit pieces to a 6-quart pot and add just enough fresh water to cover the meat. Add the other 2 tablespoons of pickling spice and bring to a boil. Then reduce the heat to low, cover, and gently simmer for about 2 hours until the meat is falling off the bone.
When the rabbit is done, remove it from the cooking liquid and transfer it to a cutting board to let cool. Shred the meat using 2 forks. Be careful to remove all small bones.
Butter one side of each slice of rye bread. On the non-butter side spread 1 tablespoon of Thousand Island dressing on each slice. Top half of the slices with cheese, corned rabbit, and sauerkraut. Top each sandwich with remaining slices, dressing side down.
Heat a large skillet over medium heat. Place a sandwich in the skillet and cook until golden and cheese is melted, about 4 minutes per side.
Lift the sandwiches onto a cutting board. Cut each in half diagonally and serve.