First, make the beet juice. Blend beet chunks and 1½ quarts of water in a blender until smooth. Strain juice and discard beet pulp. Reserve the beet liquid/juice.
In a medium to large pot, bring the remaining 3 quarts of water and 1 Tablespoon of salt to a boil. Add pasta and cook for about 2-3 minutes or until bendable. At the same time pasta is cooking, bring 4 cups (or 1 quart) of the reserved beet juice, a sprinkle of salt, and vinegar to a boil over medium high heat in a large sauté pan.
Using tongs or a pasta strainer, lift out partially cooked pasta from the pot and place in the sauté pan with the boiling juice. Cook pasta until al dente, stirring frequently, over medium-high heat. You may need to add another cup or two (or 1/4 to 1/2 quart) of the remaining beet juice to finish cooking the pasta. Once pasta is cooked, strain.
You can sauté colored, strained pasta with oil and/or butter, garlic, herbs, cheese, and pretty much anything else that you want.