8 tablespoons olive oil
1 cup (packed) fresh basil leaves
2 large shallots
4 garlic cloves olive oil for brushing pan 
1 1/2 pounds red potatoes 
1 1/2 pounds yellow potatoes 
1 pound purple potatoes
3/4 cup freshly grated Parmesan cheese

Blend 4 tablespoons olive oil, basil, shallots, and garlic in processor until smooth. Season basil sauce to taste with salt and pepper. Preheat oven to 400°F. Coat baking sheet with oil. Cut all potatoes into 1 1/2" pieces. Toss potatoes with 4 tablespoons olive oil, salt, and pepper in large bowl to coat. Transfer to prepared baking sheet. Roast potatoes until almost tender, about 35 minutes. Pour basil sauce over potatoes and toss to coat. Continue roasting potatoes until golden brown and tender when pierced with skewer, about 20 minutes longer. Transfer potatoes to serving bowl. Add cheese and toss to coat.