Cook potatoes in large pot of boiling salted water until tender, about 10 minutes. Using slotted spoon, transfer potatoes to cutting board; return water to boil. Add broccoli; cook 3 minutes. Using slotted spoon, transfer broccoli to large bowl; return water to boil. Add linguine and cook until tender but still firm to bite, stirring occasionally. Drain, reserving 1/2 cup cooking water. Cut potatoes into small chunks. Add potatoes, pasta, and pesto to broccoli and toss, adding reserved cooking water by tablespoonfuls to moisten if necessary. Season with salt and pepper. Serve, passing cheese separately.
