Peel the potatoes and cut them into cubes.
Bring a pot of water to a boil, add 3/4 teaspoon salt and when the water is boiling, add in your potatoes. Reduce heat to medium.
Boil your potatoes for approximately 15 minutes, until they are soft enough to be mashed. (The exact amount of time that this takes will depend on the size of your potato pieces). Reserve 1/4 cup of the cooking water.
While your potatoes are boiling, prepare your garlic. Peel the garlic and remove and discard any green part that you see in the individual cloves of garlic.
Combine the garlic with 1 teaspoon of salt and using a mortar and pestle turn it into a paste. If you do not have a mortar and pestle you can use a micro-grater or garlic press to prepare your garlic and then mix it together with the salt.
When the potatoes have boiled, drain and mash them. If you need to add some liquid to make them creamier and easier to mash, add in up to 1/4 cup of the water that they were boiled in.
Allow your mashed potatoes to cool.
In a stand mixer fitted with the whisk attachment combine the mashed potatoes, garlic paste, red wine vinegar and vegetable oil. Whisk together for 3 - 5 minutes on medium speed until all of the ingredients are well combined.