For Brine: Boil 1 cup water. Add salt and stir to dissolve. Remove from heat. Add remaining water and brine ingredients. Let cool completely.
For pork: Place pork loin in container and pour in brine. If brine does not cover pork completely, add more water until it is completely submerged.
Place container in refrigerator, leaving it for 8-12 hours.
Remove pork. Rinse with water and dry thoroughly on paper towels.
Mix together paprika, thyme, cayenne, salt, and pepper in small bowl.
Rub mixture all over pork.
In a sauté pan large enough to hold pork loin, heat oil over medium heat.
Sear pork on all sides, about 8 minutes total.
Remove pork from pan and put in slow cooker.
Add butter, honey, and balsamic vinegar to pan, stirring until butter has melted. Add sliced garlic. Season with salt and pepper.
Pour sauce over pork loin. Cook on low for 3.5 to 4.5 hours until desired done ness is reached.
Remove pork and cover with foil.
Pour sauce into small saucepan. Cook on low heat, simmering until slightly thickened. Re-season with salt and pepper.