Place the potatoes in a saucepan and cover with cold water, adding 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. Cover and refrigerate the potatoes until you’re ready to grill them.
Cut the potatoes down the middle and put them in a mixing bowl. Include 1 tablespoon of olive oil, as well as salt and pepper to taste. Stir everything together so that the potatoes are coated evenly.
Transfer the potatoes to a vegetable grill basket and place them on a barbecue over high heat.
Gently grill the potatoes until they are browned and heated through. Toss them every few minutes, for a cook time of 5 to 10 minutes.
To make the Lemon Herb Sauce, combine the parsley, mint, garlic, lemon juice, and salt in a food processor. Turn on the food processor and slowly drizzle in the olive oil through the funnel. Stop to scrape down the sides of the bowl as needed. The sauce should be chunky when finished.
When the potatoes are done, move them to a serving bowl. Pour the Lemon Herb Sauce over top and mix it all together. Serve right away.