Rinse the corned beef under cold water and pat dry. This helps keep the brine flavor from overpowering. Place in a large pot.
Pour in water or broth, add onion, garlic, mustard seeds, and peppercorns. Bring to a simmer, then reduce heat to maintain a gentle bubble. Cover.
Simmer for about 2.5 to 3 hours, turning the meat halfway if you’re feeling ambitious. You want it tender enough to slice easily.
Add potatoes and carrots to the pot. Continue simmering about 15 minutes, then add cabbage wedges. Cook until all vegetables are tender but not mushy, another 10–15 minutes.
Remove the meat and vegetables to a warm platter. Let the corned beef rest 5–10 minutes before slicing across the grain. This keeps every bite juicy.
Slice the brisket, arrange with vegetables, and spoon a little broth over the top. Taste and adjust salt if needed. Optional: toss a pat of butter on the cabbage for extra shine.