Rinse the excess salt mixture from the corned beef.
Place the beef in a pot. If it's too big to fit, cut it into smaller pieces and wedge them in together.
Cover the beef with at least an inch of water.
Insert a probe (RFX MEAT works great) into the beef and either set up another probe temping the water using a pot clip (or dangle a waterproof probe in the water directly). Set the high-temp alarms for both the water and the meat to 190°F (88°C).
Bring your water up to your target temperature, and maintain it there by adjusting the heat on your burner.
Monitor the internal temp of your corned beef. Once it reaches your target temperature, let the meat cook until it can be easily pierced by a knife or Thermapen ONE—at least 2 hours after the target temp is reached. You may need to add more water in the course of the cook, as some will be lost to steam. You may also want to weigh the beef down with a bowl or plate if it doesn't want to stay submerged.
Once your beef is nearly tender enough, add the vegetables to the pot and simmer them together for at least 20 minutes, or until they reach the texture you like.
Remove the beef from the pot and let it cool enough to handle.
Slice the beef to your liking, and serve with the cooked veggies and some good mustard