Use a potato peeler to pare off the skins as thinly as possible and then cut the potatoes into even-sized chunks, not too small. If they are large, quarter them and if they are small, halve them.
Put the potato chunks in a large saucepan, then pour boiling water over them, add 1 dessertspoon of salt, put on a lid and simmer gently until they are absolutely tender - they should take approximately 25 minutes. The way to tell whether they are ready is to pierce them with a skewer in the thickest part; the potato should not be hard in the centre. And you need to be careful here, because if they are slightly underdone you do get lumps.
When the potatoes are cooked, drain them. Cover them with a clean tea cloth to absorb some of the steam for about 5 minutes, then add the pesto and a seasoning of salt and pepper. Using an electric hand whisk, start whisking at a slow speed to break up the potatoes, then increase the speed to fast and whip them up to a smooth purée.
Now return the saucepan to a low heat and use a wooden spoon to turn the potatoes round until they become hot - about 1 minute - season, and now they're ready to serve.