Bring a medium pot of salted water to a boil. Add the broccoli and cook until tender, about 10 minutes. Drain thoroughly.
For the mashed potatoes:
Put the potatoes in a medium pot, cover with cold water and add 2 tablespoons salt. Bring just to a simmer and simmer the potatoes, uncovered, until tender, about 45 minutes. Drain.
Peel the potatoes and return them to the pot (if desired, pass them through a ricer or food mill after peeling). Add the broccoli. Mash the potatoes and broccoli with the milk and butter using a fork or potato masher. Stir in the chives, dill, parsley and 2 teaspoons salt; season with pepper.
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