Preheat the oven to 425F.
Wash the beets well, and then cut them into cubes. Peel the garlic cloves. Place the beets and garlic in foil, cover them with olive oil and season with salt.
Enclose the vegetables in foil and roast in the oven at 425F for 30 minutes, or until the beets are fork tender.
While the vegetables are roasting, grate the parmesan and set aside.
When the vegetables have 10 minutes of roasting time left, bring a pot of water to a boil and salt it like the sea. Cook your pasta to al dente, drain and reserve about 1 cup of the pasta water.
In a large blender, add the beets and garlic and a splash of pasta water. Blend until smooth. I recommend doing this in a blender where you can ventilate the top, to avoid any pressure build-up as you blend. Add pasta water as needed, you want this sauce to feel like a very think purée.
Keep the pasta in the pot you boiled it in, add a generous splash of pasta water, the beet sauce and a handful of the parmesan. Stir until glossy. Continue to add handfuls of parmesan and splashes of water until you reach your desired sauce thickness and texture. Reserve at least 1/4 cup parmesan for topping.
Plate and top with leftover parmesan, salt and pepper. Enjoy immediately :)