Melt 2 tablespoons of the butter in a 10-inch cast-iron skillet over medium heat. Add the celery, carrots, leeks and tarragon, and cook, stirring occasionally, until the vegetables are softened, 3 to 5 minutes. Transfer to a bowl and set aside. Wipe out the skillet.
Add the remaining 4 tablespoons butter to the skillet over medium heat, sprinkle in the flour and whisk until a paste forms and starts to bubble. Whisk in the chicken broth and continue to cook until it begins to thicken, 3 to 5 minutes. Whisk in the cream and cook until it thickens, 1 to 2 minutes. Stir in the salt and pepper.
Return the vegetables to the skillet; add the peas and chicken and stir to combine. Sprinkle the frozen tater tots all over the filling to cover completely (there will be extra tater tots; reserve for another use). Bake until the tots are golden brown and the filling is bubbling, about 30 minutes. Let cool for 5 minutes before serving.