For the parsley pesto: Bring a medium pot of water to a rolling boil. Add the kosher salt until the water tastes like seawater and stir to blend. Add the parsley and cook for 1 minute, and then remove with a slotted spoon. Immediately plunge leaves into an ice bath and allow to cool, 3 to 5 minutes. Drain the leaves slightly and then put them in the blender with the sugar and olive oil. Blend until smooth. If the mixture is too thick, add some water to facilitate blending. Taste for seasoning. Stir in the scallions and set aside in a bowl large enough to hold the pasta.
For the beet vinaigrette: Put the beet pieces in a medium saucepan and add 1 cup of water. Bring to a rolling boil over high heat. Allow it to reduce until there is about 1/4 cup liquid. Add the remaining 1 cup of water and reduce again until you have only 1/4 cup left. Season to taste. Remove from the heat and stir in the red wine vinegar and olive oil.
For the pasta: In a large pot, bring 6 quarts of water to a rolling boil. Add a generous amount of salt; the pasta water should taste like seawater. Add the pasta to the pot and stir so it doesn't stick to the bottom as it cooks. Cook the pasta until al dente (chewy but not hard or raw tasting) 8 to 10 minutes. Drain the pasta in a colander, reserving some pasta cooking liquid in case you need it later.
To serve: Pour the pasta into the bowl with the parsley pesto. Add the beet vinaigrette and butter and toss to blend. Add the Parmesan and finely sliced beet greens, if using. Serve immediately. Alternatively, you can drizzle the red beet vinaigrette over the pasta for a different color effect.