When the internal temperature of the meat reaches 143 F, remove the meat from the oven and the cast iron pan and place on a plate or tray and cover with foil or a lid. Set aside. Leave any juices in the cooking pan.
Place your pan back on the stove and turn the heat to medium. Add the shallots and garlic and cook those for a couple mins. Then add the tea, balsamic vinegar, sugar, salt, and pepper and stir for a few mins while it simmers, scraping any bits off the bottom of the pan.
When the sauce has reduced slightly and has simmered for a few mins, add the knob of butter. Start with 1 tbsp and let it melt. Taste it and if its good, you are done. If you think it could use a little more silkiness add a little more at a time until it tastes good.
When your happy with the sauce. Unveil the tenderloin and place it on a cutting board. (You could add any juices from the tenderloin back into your sauce and give it a stir) Slice it in ½ inch - 1 inch pieces and transfer to your serving platter. Pour the sauce over and serve!