Preheat an oven to 350 degrees F (175 degrees C). Grease a 2 quart casserole dish.
Place the carrot and turnip cubes into a large pot and cover with salted water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain and allow to steam dry for a minute or two.
Return the carrots and turnips to the pot, and mash well with the butter, salt, flour and 1/4 cup of sugar. Whisk together the eggs and milk in a bowl, then stir into the mashed carrot mixture until blended. Scrape into the prepared casserole dish. Stir together 2 tablespoons of sugar with the cinnamon in a small bowl, and sprinkle over the top of the carrot mash.
Bake in the preheated oven until the carrot mash has slightly firmed and the top has begun to turn golden brown, about 45 minutes.