Preheat oven to 350 F. Remove the silver skin from the pork and sprinkle on all sides with kosher salt and freshly ground black pepper.
In a large cast-iron skillet, heat 2 tablespoons cooking oil over high heat. Add the pork tenderloins and brown on both sides.
Transfer the skillet to the oven. (If you don't have an oven-proof skillet, transfer the pork to a baking dish. Don't wash out the skillet though, you will need it later for the sauce.)
Bake the pork until a meat thermometer registers 140 F. (For me this is about 12-15 minutes normally.) Take the pork out of the oven and transfer to a cutting board to rest.
Pour off any of the juice from the skillet (or baking dish) into a measuring cup. Add enough apple cider to make 1-1/2 cups of liquid.
Set the skillet over medium heat. Add a few drops of oil, then the ginger. Scrape the browned bits from the pan.
When the ginger becomes fragrant, pour in the liquid and increase the heat to medium-high. Continue scraping the browned bits and bring the mixture to a simmer. Continue simmering to reduce the liquid by half.
When the liquid has thickened, add salt and pepper to taste. Add the lemon juice. Remove from heat and swirl in 2 tablespoons of butter.
Slice the pork and transfer to a serving dish. Pour the sauce over it.