Put the potatoes and salt in a large pot (at least 5 quarts). Cover with water and bring to a boil.
Lower the heat and partially cover the pot. Cook until the potatoes are tender when pierced with the tip of a knife, 20 - 25 minutes. Throw the peas into the pot for the last 15 seconds of cooking. Drain and cool the potatoes 15 minutes.
Stir together the aioli, tarragon and a good 15-20 grinds of pepper.
Transfer the potatoes to a serving bowl. Using the back of a large wooden spoon, press down on the potatoes to lightly smoosh and crack them. Add 3/4 cup of the aioli to the potatoes and toss gently to coat. Taste and add more aioli if you like.
Aioli will keep up to 5 days in the refrigerator. Mix the salad just before serving.