Using a fork, poke holes in the potatoes. Place them in a large pot and fill the pot with enough cold water to cover the potatoes. On high heat, bring to a boil. Then reduce the heat to medium and simmer the potatoes until they are soft, tender, and easily pierced by a fork or skewer.
While the potatoes cook, add the olive oil to a large saute pan on medium heat. Add the diced onion and cook until the onion is beginning to become translucent, 2-3 minutes. Add the garlic and cook until it becomes aromatic, another 1-2 minutes. Then add the tomato paste and cook, mixing together, until the tomato paste begins to caramelize, 1-2 minutes. If you are adding red pepper flakes, add them with the tomato paste.
Once the tomato paste has begun to caramelize, add the can of tomatoes and the vodka. Crush the whole tomatoes little with a fork while bringing to a rapid simmer on medium-high heat. Once the sauce begins to bubble, reduce the heat to medium-low. Add the balsamic vinegar and salt and black pepper to taste, then cover the pan and let simmer 15-20 minutes.
After the sauce has simmered, taste it. If it is too acidic, add sugar little by little to taste. Add more salt and pepper to taste, then turn off the heat. Ladle the sauce into the food processor, add the basil, and process it until smooth, processing in batches if necessary. Return the sauce to the saute pan (heat off).