Bring a large pot of salted water to a boil. Add edamame and cook until tender, 3 or 4 minutes. Using a slotted spoon, scoop into a bowl. Add pasta to water and cook until just tender, about 10 minutes.
While pasta cooks, steam squash in a separate pot in a steamer basket set over boiling water, covered, until very soft and cooked through, 12 to 13 minutes. Scoop squash straight into a blender or food processor. Add cheese, butter, and 3/4 cup milk; blitz into a smooth, thick, but still pourable sauce. If it seems too thick, add a little extra milk. Season with the salt and pepper.
Drain pasta, return to empty pasta pot, and pour in butternut squash sauce; mix. Spoon into a serving dish and top with edamame.