Place the whole potatoes in a saucepan and cover with cold water. Add about a teaspoon of salt. Bring to a boil, then reduce the heat and simmer, uncovered, until the potatoes are tender when pierced with a fork. This step can be done in advance, even the day before. Cover and refrigerate the potatoes until you're ready to grill them.
Slice the potatoes in half and place them in a mixing bowl. Add 1 tablespoon of olive oil and salt and pepper. Toss well to coat.
Transfer the potatoes to a vegetable grill basket. Place on a grill over high heat.
Grill the potatoes until lightly browned and heated through, tossing occasionally, about 5 to 10 minutes.
Meanwhile, prepare the Lemon Herb Sauce by placing the parsley, mint, garlic, lemon juice, and salt in the bowl of a food processor. Process while drizzling the olive oil through the funnel of the food processor. Scrape down the sides with a spatula as needed. The sauce should be chunky.
When the potatoes are ready, transfer them to a serving bowl. Top with the Lemon Herb Sauce and toss to coat. Serve immediately.