For the potatoes: Place the potatoes in a large pot of water seasoned with a couple pinches of salt. Bring to a simmer and cook until the potatoes are soft, 7 to 8 minutes.
Meanwhile, put the cream, butter and thyme in a medium saucepan. Shave 4 to 5 slices from the truffle and add them and any broken-up parts to the cream mixture. Bring to simmer over medium heat, then cook on low to infuse the cream.
Strain the potatoes. Put them through a ricer and back into the pot. Finely grate the remaining truffle on a rasp grater into the potatoes. Gradually strain the infused heavy cream into the mashed potatoes. Season with salt and a drizzle of olive oil. Stir well with a wire whisk.
For the steak: Preheat the oven to 400 degrees F.
Dry the steak off on both sides very well and sprinkle liberally with salt and pepper.
Heat the olive oil in a large saute pan over high heat. Once the oil starts to smoke, sear the steak until a nice brown crust forms, 2 to 3 minutes per side.
Add the shallots cut-side down in the residual fat in the pan and cook to caramelization, 3 to 4 minutes. Remove the shallots and steak and drain the fat from the pan. Return the steak to the pan over medium heat. Add the butter and thyme and cook until the thyme starts to pop and fry. Baste the steak with the butter and thyme mixture. Add the shallots back to the pan and Finish with the lemon juice.
Transfer the steak to a baking sheet lined with a rack and smother in the caramelized shallots, butter and thyme mixture. (Reserve the pan it was cooked in to make the sauce.) Transfer the baking sheet to the oven and roast until medium-rare and an instant-read thermometer inserted in the center (do not touch bone) reads 132 degrees F, about 15 minutes. Let the steak rest before slicing.
For the sauce: Add the olive oil, shallots and thyme to the drippings in the pan and saute over medium heat until the shallots are translucent, 3 to 4 minutes. Deglaze the pan with the red wine, then turn the heat down to medium and reduce the wine by two-thirds, about 8 minutes. Add the demi-glace and red wine vinegar and season with salt and pepper.
Whisk in the cold butter in, one piece at a time, to mount the sauce. Strain the reduction, removing the shallots and thyme. Return the reduction to the pan and warm over low heat.
Serve the potatoes and steak with the beurre rouge. Garnish with the reserved caramelized shallots and some watercress.