In the bowl of a food processor, puree the basil, pine nuts, garlic, and balsamic until finely chopped. With the processor running, add the oil in a stream until it is all incorporated. Scrape the pesto out of the processor and into a bowl. Add the parmesan cheese and mix it in. Taste and add salt and pepper as needed.
Everything cooks in the same pot of water. Don't dump out the water until the very end.
Fill a large pot with water. Add the potatoes and a couple of tablespoons of kosher salt. Bring to a boil over high heat. Once the potatoes have begun to boil, cook them for 12-15 minutes, or until a knife easily pierces the potatoes. Use a spider or slotted spoon (or a mesh strainer) to scoop out the potatoes into a bowl.
Add the green beans to the pot and cook them for 5-6 minutes, or until tender. Scoop them out into the bowl.