How to Make It
Bring a small pot of salted water to a boil. Add the chopped carrot and celery root and cook over high heat until just tender, 4 to 5 minutes. Drain and let cool under running water; drain again. Transfer the vegetables to a plate lined with paper towels and pat dry.
Melt the butter in a large saucepan. Add the onion and cook over moderate heat until softened, about 5 minutes. Add 2/3 cup of the flour and cook, stirring, for 1 minute. Transfer the paste to a large heatproof bowl and slowly whisk in the milk to form a smooth slurry. Return the mixture to the saucepan and cook over moderate heat, stirring, until bubbling and very thick, about 6 minutes. Add the tuna to the white sauce along with the carrots, celery root, salt and pepper. Scrape the croquette mixture into a lightly oiled medium bowl and refrigerate until chilled and firm, at least 4 hours, or preferably overnight.
Spread the remaining 1 1/3 cups of flour in a shallow bowl or pie plate. Put the eggs and bread crumbs in 2 shallow bowls or pie plates. Using 2 spoons, shape slightly rounded tablespoons of the croquette mixture into oval patties and drop them, 3 or 4 at a time, into the flour. Turn the croquettes to gently coat them with flour, then dip them in the eggs and coat with bread crumbs. Return the croquettes to the eggs and then the bread crumbs to give them a light second coating. Transfer the finished croquettes to a baking sheet lined with wax paper and continue shaping and coating the remaining croquettes. Loosely cover with wax paper and refrigerate until chilled, at least 30 minutes.
Heat 1 inch of vegetable oil in a large deep skillet until shimmering. Working in batches, cook the croquettes over moderately high heat, until golden and crisp, turning frequently, about 2 minutes; lower the heat if they brown too quickly. Using a slotted spoon, transfer the croquettes to paper towels to drain. Serve warm or at room temperature.