MAKE THE SWEET ONIONS: Heat 2.5 tablespoons of the butter in a heavy skillet on medium heat. Add the onions to the skillet along with a bit of salt. Stir occasionally, but not too much. Cook for approximately 30 minutes to get the onions to a nice deep brown color.
PREPARE THE PORK: As the onions get close to being ready, begin prepping the pork. Slice the pork tenderloin crosswise into medallions about 1 inch thick. Lay slices on work surface. Season with salt and white pepper.
Melt the remaining butter in a large skillet, preferably oven proof. Depending on size of your skillet and amount of pork, you may need two skillets or to cook in two batches.
When the butter has melted, add the pork medallions and cook for 3 minutes to brown one side. Turn the pork over, and cook for another 3 minutes.
Move the pork to the oven, and let rest at the low heat of 200 degrees for 10 minutes.
Check the pork for doneness after 10 minutes. Pork should be cooked to 145 degrees; pork tenderloin should still have some pink to be ender and moist. Put the pork back in the oven for a few more minutes to finish if needed.
When the pork is ready, plate several medallions and serve the onions over or alongside the pork. Enjoy!