Place turkey breast in a large pot and add broth, leeks, turnips and herbs. Bring to a boil, lower heat, and simmer, covered, until turkey is cooked through and vegetables are tender, 35 to 40 minutes.
Meanwhile, boil potatoes and celery root in separate pots covered by a few inches of cold water until very tender, about 15 minutes. Drain, combine in one pot and add almond milk. Mash with a potato masher or large fork until smooth.
Remove turkey from broth and set on a platter. Discard herbs. With a slotted spoon, spoon leeks and turnips around turkey and keep warm. Return pot with broth to medium-high heat and boil until the liquid is reduced to about 3/4 cup; pour over and around turkey. Reheat vegetable mash; if necessary, and carve turkey into thin slices.
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