For the topping: Place the celery root in a large pot and add 2 cups chicken broth, along with water to cover and 2 teaspoons salt. Bring to a boil over high heat, then reduce the heat to medium and simmer until the celery root is tender, 15 to 20 minutes.
Meanwhile, make the filling: Heat the olive oil in a large skillet over medium heat. Add the onions, carrots and celery and season with a teaspoon of salt and a few grinds of pepper. Cook until the vegetables are tender, about 5 minutes. Add the garlic and thyme and cook until fragrant, another minute. Add the ground turkey and cook, breaking up with a wooden spoon, until browned, about 5 minutes. Stir in the flour until incorporated, about 1 minute. Add the remaining cup chicken broth and reduce the heat to a simmer; cook until thickened, 3 to 5 minutes.
To finish the topping, reserve 1 cup of the cooking liquid, then drain the cooked celery root and set aside. To the pot, add the butter and melt over medium heat. Return the celery root to the pot and add half of the reserved cooking liquid. Using an immersion blender, puree the mixture until smooth, adding more of the reserved liquid as necessary. (Note: If you don't have an immersion blender, transfer the mixture to a food processor and pulse until the mixture is pureed and smooth.) Season with salt and pepper.
Transfer the filling to a 9-inch square baking dish. Sprinkle the frozen peas in an even layer over the filling. Spread the celery root puree on top, being careful to leave the layer of peas in place.
Bake until the topping is lightly golden and the filling is bubbling, 30 to 35 minutes.