Heat the oven to 450 degrees F. In a large bowl, toss the potatoes with the rosemary, 2 tablespoons of the olive oil, 1 teaspoon salt, and a few generous grinds of pepper. Rub the remaining 1 tablespoon olive oil on a heavy-duty rimmed baking sheet. Spread the potatoes on the baking sheet and roast, tossing after 15 minutes, until they're nicely browned and tender when pierced with a paring knife, about 30 minutes.

While the potatoes are still hot, toss with 1/2 cup of the pesto, the scallions, and the mint in a large bowl. Transfer to a serving platter, garnish with mint sprigs if you like, and serve warm.

Put the olives, olive oil, mint, shallot, and lemon juice in a food processor. Process, stopping to scrape the sides of the bowl with a rubber spatula after a couple of seconds, until the mixture is smooth. Season the pesto to taste with lemon juice, salt, and pepper.