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Place the potatoes and parsnips in a pot and cover with water. Salt the water, bring to a boil and cook to tender. Drain and return the vegetables to the hot pot. Mash with the cream cheese, milk, chives, horseradish, some salt and pepper and the egg. Cover and reserve in the pot.
Put a pot on the stove with 1/4 inch of water and bring to a boil. Remove the stem of the head of cauliflower and set it into the pot. Scatter the trimmed broccolini around and season with salt. Cover and cook to tender. Break the cauliflower into florets and drain. Remove with a slotted spoon and set aside.
Place a cooling rack over a baking sheet, arrange the bread on it and toast in the oven. Coarsely chop the bread, add it to a food processor and process into coarse breadcrumbs. Melt the butter in a saucepot over medium heat. Pour half of the butter over the crumbs to moisten them.
To the remaining butter, add the garlic and gently cook 2 minutes, stirring constantly. Add the hot sauce and mustard and stir to combine. Sprinkle in the flour and cook 1 minute. Whisk in the beer, Worcestershire sauce and lemon juice and cook the beer out for 30 seconds. Add the stock, bring to a bubble and thicken a minute more. Stir in the cheese in a figure-eight motion and season with salt and pepper.
Grease a casserole dish. Layer the potatoes and parsnips, cauliflower and broccolini, cheese sauce and breadcrumbs in the dish. Cool and store for a make-ahead meal, or bake to serve. Bake at 375 degrees F from room temperature until bubbly and golden brown, about 50 minutes.