Make the corned beef: Put beef, carrots, and chopped onions into a large pot, cover with water, and bring to a boil over high heat. Reduce heat to medium-low, skim foam, cover, and simmer until tender, about 2 1⁄2 hours. Reserve 3⁄4 cup of the cooking liquid.
Make the champ (mashed potatoes): Put potatoes into a steamer basket set over a large pot of boiling water, cover, and steam over medium heat until tender, 1 hour. Put milk, scallions, and 4 tbsp. butter into a small pot and simmer over medium heat until scallions have softened, about 1 minute. Mash potatoes and milk mixture in a bowl with a potato masher until fluffy. Season with salt and pepper.
For the carrot-parsnip mash: Put carrots and parsnips into a medium pot, cover with salted water, and bring to a boil over medium-high heat. Reduce heat to medium-low and simmer, partially covered, until carrots and parsnips are tender, 30–35 minutes. Drain, transfer to a bowl, and mash with 2 tbsp. butter and parsley. Season to taste with salt and pepper and set aside.
Make the broccoli: Bring a large pot of salted to water to a boil over high heat. Blanch broccoli spears in batches until just tender, 3-5 minutes, transferring when done to an ice bath to cool. Drain well and set aside.
Make the parsley sauce: Melt butter in a small saucepan over medium heat. Add minced onions and cook until just softened, about 1 minute. Whisk in flour and cook until light golden, about 1 minute. Whisk in reserved beef cooking liquid, milk, parsley, mustard, and nutmeg and whisk until smooth. Season to taste with salt and pepper and cook, whisking, until thickened, 4–5 minutes.
To serve, reheat champ and carrot-parsnip mash in separate pots. Melt 2 tbsp. butter in a large skillet, add broccoli, and heat through, then season. Slice corned beef and serve with parsley sauce, champ, carrot-parsnip mash, and broccoli.