Using food processor grate the beets and set them aside in a large serving bowl.

Put the poppy seeds into a large saucepan set over over medium-high heat. Toast the seeds about 3 minutes until they are fragrant, stirring constantly. Transfer them to a small bowl and set them aside. Using the same saucepan add 4 tablespoons butter and cook over medium heat. Once it begins to brown remove it from the heat and pour it over the raw beats. Season with salt and pepper and the balsamic vinegar.

Meanwhile, in a large pot of salted water, boil the pasta until al dente. Drain the pasta reserving about 1-cup of the pasta water. Toss the pasta with the remaining 2 tablespoons of butter, stirring until melted. Follow the butter with the grated cheese, reserved poppy seeds and the beets; then add the reserved pasta water and stir quite well. Adding the butter first keeps the cheese from clumping. The heat from the pasta is all the cooking necessary to melt the cheese and cook the beets. Add the reserved poppy seeds. Garnish with mint. Serve with a little more cheese passed at the table.