Prepare a thermocirculator and water bath according to manufacturer's directions. Set thermocirculator to 135 degrees F (57 degrees C) and allow water bath to come up to temperature. Season rib roast evenly with 1 tablespoon salt and transfer to a large vacuum-sealable bag; arrange butter cubes, thyme sprigs, and crushed garlic around rib roast in bag.

Seal bag with a vacuum sealer according to manufacturer's directions. Transfer vacuum-sealed rib roast to warm water bath with thermocirculator, cover vessel with aluminum foil to reduce evaporation, and cook for 6 hours. Remove rib roast from bag; roast will be at 135 degrees F (57 degrees C) when pulled from water bath. Transfer to a rimmed sheet pan lined with paper towels and gently pat dry; reserve contents remaining in bag.

Preheat the oven to broil with an oven rack positioned about 10 inches below the heat source. Line a rimmed baking sheet with aluminum foil and fit with a wire rack.

Season rib roast evenly with remaining tablespoon salt and pepper and place on the prepared baking pan.

Cook rib roast in the preheated oven until fat cap is deeply caramelized, 8 to 10 minutes. Transfer rib roast to a large cutting board and let rest for 10 minutes before slicing against the grain into 1- to 1 1/2-inch thick slices; garnish sliced prime rib with flaky sea salt to taste.

While rib roast is in the oven, combine reserved contents from vacuum bag and mushrooms in a medium saucepan and cook over medium-high heat, stirring occasionally, until mushrooms are mostly tender and liquid has reduced completely, 8 to 10 minutes. Add wine and continue cooking, stirring occasionally, until wine is almost completely reduced, about 5 minutes. Add beef broth and continue cooking, stirring occasionally, until sauce is thick enough to coat the back of a spoon, about 15 minutes; remove thyme sprigs and discard. Season mushroom sauce to taste with salt and serve alongside sliced prime rib.