Combine the potatoes, celery root, milk, 1 teaspoon salt and a few grinds of pepper in a pot. Bring to a boil, then reduce the heat to low. Cover and cook until the vegetables are soft but not falling apart, about 15 minutes. Stir in the kale and scallions; cover and cook, stirring occasionally, until the kale is tender, about 6 minutes.
Set a colander over a bowl and drain the vegetables, reserving the milk. Return the vegetables to the pot and add 4 tablespoons butter and 1/2 cup of the reserved milk. Mash to combine, adding up to 1/4 cup more milk if needed.
Preheat the oven to 400 degrees F. Melt the remaining 2 tablespoons butter in a large pot over medium heat. Add the shallots, carrots, 1 teaspoon salt and a few grinds of pepper. Cook, stirring, until the vegetables are soft, about 15 minutes. Sprinkle with the flour and cook, stirring, 1 minute. Stir in the chicken broth. Bring to a boil, then reduce the heat to medium low; stir in the chicken and simmer until just cooked through, about 8 minutes. Remove from the heat and stir in the chives and parsley.
Divide the chicken mixture among six 12-ounce casserole dishes or ramekins. Dollop the mashed potatoes on top, then spread with the back of a spoon. Bake until bubbling around the edges and the topping is browned in spots, about 20 minutes. Let rest 5 minutes before serving.
You can also make this shepherd's pie in a 3-quart baking dish.