Boil the potatoes: Add the potatoes to a medium pot with the cream, garlic, butter and some kosher salt. Bring to a boil over medium-high heat, then reduce the heat to medium-low and gently simmer until the potatoes are easily pierced with a fork, about 25 minutes.
Cook the chateaubriand: Preheat the oven to 350 degrees F. Let the beef rest at room temperature for 15 minutes, then season with salt and pepper. Heat the olive oil in a large, oven-safe skillet over high heat until it shimmers, about 2 minutes. Set the meat in the pan and sear it on all sides until browned, about 8 minutes total. Reduce the heat to low and add 2 tablespoons of the butter, the mushrooms, garlic and the thyme. Transfer the skillet to the oven and cook the roast until an instant-read thermometer inserted into the thickest part reads 125 to 130 degrees F., 6 to 8 minutes for medium-rare. Remove the skillet from the oven, transfer the roast to a cutting board, loosely tent with foil and let it rest for 10 minutes.
Make the red wine reduction: Transfer the mushrooms to a bowl and pour off all but 1 tablespoon of the fat from the pan (discard the garlic and thyme). Set the skillet over medium-high heat, add the shallot and red wine, and cook, stirring to scrape up any browned bits from the bottom of the pan, until the pan is almost dry, about 2 minutes. Add the broth and cream, bring to a boil, and simmer until the sauce thickens, 3 to 5 minutes. Return the mushrooms to the pan, adjust the seasoning with salt and pepper, if needed.
Make the pomme puree: Place a colander over a large bowl. Pour the hot potatoes into the colander. Shake the potatoes to drain off the excess liquid. Press the potatoes through a ricer (or mash them with a fork for a chunkier texture) and into a medium bowl, occasionally clearing the sides of the ricer with a spoon. Pour the reserved and still-hot cream over the potatoes in small increments, folding in to combine, until you get your desired consistency. Season with olive oil and a sprinkle of sea salt and serve immediately, or keep warm in a covered oven-safe casserole dish in a 250 degree F. oven.
To serve, remove the twine from the roast and slice the roast crosswise into 2 equal portions. Sprinkle with salt and pepper. Spoon the potato puree onto 2 warm plates. Place a slice of chateaubriand on each plate and spoon the red wine-mushroom reduction over the top. Garnish with fresh thyme leaves and serve.