Make the pesto and set aside.
Bring a large pot of salted water to a rolling boil.
Add the potatoes and green beans and cook until tender, about 7 minutes.
Using a large slotted spoon, transfer the potatoes and beans to a large serving bowl.
Cover the bowl lightly with foil to keep the veggies warm.
Bring the water back to a rolling boil and add the penne, stir well, and cook, stirring occasionally, until al dente, following package instructions.
Scoop out and reserve about 2 cups of the pasta water, then drain the pasta.
Add the drained pasta to the bowl with the veggies and then add the pesto.
Stir and toss until the pasta and veggies are evenly coated with the sauce, adjusting the consistency with some of the cooking water if needed.
Add the butter and gently toss to coat evenly.