Put corned beef in a pot with onion and celery and sprinkle spice packet over top. Pour in cold water to cover. Turn heat to high and bring to a boil. Skim off and discard any foam if desired. Once boiling, reduce heat to low, cover, and simmer gently until tender, about 3 1/2 hours. Ensure internal temperature reaches 145°F (63°C).

Carefully remove corned beef to a bowl and let cool. Taste broth and add salt if needed.

Bring broth to a boil over high heat. Add cabbage to the boiling broth. Cook until cabbage just starts to soften, about 5 minutes. Remove cabbage with a slotted spoon to a bowl.

Add carrots to the boiling broth. Cook until carrots start to soften, about 5 minutes. Remove carrots with a slotted spoon to a bowl.

Add potatoes to the boiling broth and reduce heat to medium. Cook until tender, 15 to 20 minutes, then remove to a bowl.

While potatoes cook, slice the cooled corned beef along the fatty seam to separate into 2 pieces. Cut across the grain for 1/4-inch slices, removing large pieces of fat.

Preheat the oven to 400°F (200°C). Butter a 9x15-inch casserole dish.

Add butter, salt, pepper, cayenne, and milk to the potatoes and mash until smooth. Stir in 3/4 cup Cheddar cheese.

Layer the casserole: compact cabbage at the bottom, followed by carrots, then corned beef slices. Pour 1 cup reserved broth over beef. Top with mashed potatoes, smoothing over with a spatula, and create a fork texture.

Bake in the center of the oven until browned and bubbling, 45 minutes to 1 hour.