Cut the leek in half lengthwise and rinse thoroughly to remove any grit, then thinly slice crosswise (you should have a tleast 2 cups). In a medium saucepan, melt 3 tbsp. butter over medium heat. Add the leeks, then stir in 1 tsp. salt and 1/2 tsp. pepper. Cover and cook over medium heat (lower the heat if the leeks are browniing), stirring, until tender, about 7 minutes; let cool.
Preheat the oven to 400 degrees and lightly grease a 9-inch square baking dish. Using a food processor fitted with the slicing disk, slice the potatoes, then the beets. Rinse the potato slices with cold water and pat dry on a kitchen towel. Transfer the potatoes and beets to a large bowl and toss to mix. Arrange one-third evenly in the prepared baking dish. Spoon half of the leeks evenly on top and sprinkle with 2 tbsp. flour. Make another layer with half of the remaining potatoes and beets and the remaining leeks and 2 tbsp. flour. Top with the remaining potatoes and beets.
In a glass measuring cup, mix the broth and half-and-half; pour evenly over the casserole. Cover snugly with foil and bake until the vegetables are tender, about 1 hour 15 minutes.
Melt the remaining 2 tsp. butter and toss with the bread cubes. Crumble the goat cheese and sprinkle on top of the gratin along with the bread cubes. Bake uncovered until golden, about 15 minutes. Let stand 5 to 10 minutes before serving.