Place potatoes in a medium saucepan with enough cold water to cover the potatoes. Bring to a boil over high heat. Reduce heat to medium-low.
Add salt, partially cover, and simmer until tender, about 10-15 minutes. Drain well.
Once completely cooled, cut potatoes in half or quarters depending on size.
In a small food processor, place the garlic, almonds, parmesan, arugula, and mint. Pulse several times. Add the oil, vinegar, and salt and pepper to taste; pulse several times again. The pesto should be fairly chunky, not too smooth.
Transfer the potatoes to a serving bowl, fold in the pesto. Cover and chill for 1-2 hours
Taste for seasoning and adjust as needed. Serve cold or at room temperature.