Preheat the oven to 350°F. Roast 2 of the beets until tender, about 1 hour. The beets can be roasted a day in advance.
When the beets are cool enough to handle, slip off the skins. Cut the beets into ⅛-inch-thick strips 2 inches long. Put the beet strips into a bowl and add 1 teaspoon of the vinegar, 2 tablespoons of the extra virgin olive oil, a pinch of sugar, and salt and pepper to taste. Arrange in a mound in the center of each of 4 large white plates.
Trim and peel the remaining beet and cut into 1-inch cubes. Put beet cubes in a food processor until coarsely pureed, about 2 minutes. Let the beet mixture sit for 10 minutes, then strain through a sieve into a small bowl; discard any beet remaining in the sieve. Stir in 1 teaspoon of the vinegar and the remaining 2 tablespoons extra virgin olive oil. Season with salt and pepper to taste. Set the beet vinaigrette aside.
Heat the olive oil in a frying pan large enough to hold all the squid in a single layer (or do in 2 batches). Season the squid and cook them in the hot oil over medium-high heat for about 30 seconds on each side. Do not be tempted to cook the squid longer, or it will become tough. Then pour the remaining vinegar into the pan and let it evaporate.
Arrange the squid squares around the beet salad. Pour some of the raw beet vinaigrette sauce around each plate.