Bring a medium pot of salted water to a boil. Peel and medium-dice the potatoes. Add the potatoes to the boiling water and cook, 14 to 16 minutes, until very tender when pierced with a fork. Reserving ½ cup cooking water (double for 4 portions), drain the potatoes and return to the pot. Off the heat, add 1 tbsp butter (double for 4 portions). Using a fork (or potato masher), mash the mixture, gradually adding the reserved cooking water until you achieve your desired consistency. Season with S&P to taste. Set aside in a warm spot.
While the mash cooks, peel, halve and thinly slice the onion. In a medium, oven-safe pan, heat a drizzle of oil on medium-high. Add the onion to the pan and cook, stirring occasionally, 5 to 7 minutes, until lightly browned. Reduce the heat to medium and add 1 tbsp water (double for 4 portions); season with ⅓ of the spice blend and S&P. Cook, stirring frequently, 2 to 4 minutes, until very tender. Stir in an additional 1 tbsp water (double for 4 portions), scraping up any browned bits (or fond) from the bottom of the pan, 1 to 2 minutes, until reduced.
While the onions caramelize, preheat the oven to broil. Trim off and discard the stem ends of the string beans. Mince the garlic. Finely chop the parsley leaves and stems.
In a large pan, heat a drizzle of oil on medium-high. Add the string beans and cook, stirring frequently, 1 to 2 minutes, until beginning to soften. Add ½ the garlic and 1 tbsp water (double for 4 portions). Cook, stirring frequently, 1 to 2 minutes, until crisp-tender; season with ½ the remaining spice blend. Transfer to a plate and set aside in a warm spot. Wipe out and reserve the pan.
In the same pan, heat a generous drizzle of oil on high. Pat the beef strips dry with paper towel; season with the remaining spice blend and S&P. Add the seasoned beef* to the pan and cook, 1 to 2 minutes per side, until cooked through.
To the pan of caramelized onions, add the demi-glace, white wine vinegar, remaining garlic and 2 tbsp water (double for 4 portions). Cook, stirring frequently, 2 to 3 minutes, until thickened. Add the cooked beef and 1 tbsp butter (double for 4 portions); stir to combine and top with the Emmental. Transfer the pan to the oven and broil, 3 to 4 minutes, until golden and bubbly. Divide the mash between your plates. Top with the finished beef and blistered string beans. Garnish with the parsley. Bon appétit!