Preheat the fryer. Preheat the oven to 400 degrees F.
In a mixing bowl, toss the fennel with the olive oil, salt and pepper. Place on a small baking sheet and roast until caramelized, about 30 minutes. Remove from the oven and cool.
Place the beets in a saucepan and cover with water. Season with salt and pepper. Bring the liquid to a boil, reduce the heat to medium and cook until tender, about 8 to 10 minutes. Drain and cool in an ice bath. Remove and drain on paper towels. Slice the beets in half.
In another shallow bowl, toss the beets with olive oil, salt and pepper. Set aside.
Place the potatoes in a saucepan, over medium-high heat and cover with water. Season with salt. Bring the water to a boil, reduce heat the medium and simmer until tender, about 4 to 6 minutes. Drain and cool in an ice bath. Remove and dry on paper towels. Toss the potatoes with olive oil, salt and pepper. Set aside.
In a mixing bowl, combine the mayonnaise, mustard, lemon juice, onions, horseradish, garlic, parsley, and tarragon. Mix until fully incorporated. Season with salt and pepper. Add the crabmeat and mix well. Season with salt and pepper.
Fry the celery root slices until golden brown. Remove and drain on paper towels. Season with salt and pepper.
To serve, fan the slices of potatoes in the center of each serving plate. Mound the crab salad in the center of the potatoes. Place the beets and fennel around the salad. Garnish the top of the crab salad with the fried celery root chips. Garnish with parsley.
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