Place potatoes in a saucepan and cover with cold water and some salt. Cover and bring to a boil. Remove lid from potatoes and simmer until potatoes can be easily pierced with a fork, 12 to 15 minutes.
Meanwhile, heat a large skillet with cooking oil over medium heat. Add pork and sear on all sides until deep golden brown, 8 to 10 minutes total.
Transfer pork to a sheet pan. (Leave the pan you used to brown the pork on the stove, but away from the heat source.)
Toss green beans with olive oil, mustard (portion for green beans), and balsamic vinegar. Season with some salt and pepper and spread out around pork.
Transfer pork and green beans to the oven and roast until vegetables are tender and pork registers 145F / 63C, 13 to 18 minutes.
Meanwhile, return the pan you used to sear the pork to medium heat. Add butter (portion for pork). To melted butter, add shallots and tarragon, scraping up any browned bits on the bottom of the pan. Saute until shallots are soft, 2 to 3 minutes. Whisk in apple cider vinegar and mustard (portion for pork). Stir in flour until no dry spots remain. Slowly pour in stock while whisking. Simmer sauce until thickened, 2 to 3 minutes. Season with some salt and pepper.
Drain potatoes and let sit in a colander for ~5 minutes to allow the steam to escape (this creates creamier potatoes vs. liquidy potatoes).
Use a masher to mash potatoes with sour cream, butter (portion for potatoes), and some salt.
Transfer cooked pork to a cutting board and rest, covered, for 5 minutes.
When pork has rested, cut into slices.
Serve pork over mashed potatoes with pan sauce drizzled over both. Enjoy green beans on the side.