Pour stock in a large saucepan, then add the scallions.
Cook over low heat, stirring frequently until the scallions are limp.
Stir in the parsley, and when it is wilted, add the flour.
Stir until the flour coats the scallions, then add the garlic by pushing it through a garlic press into the saucepan.
Add the milk a little at a time, stirring to break up any lumps of flour.
Turn heat to medium.
Cut the florets from the broccoli, making them about 1 1/2 inches long.
Peel any tough outer skin from the stems of the florets.
Cut the florets in thin, lengthwise strips.
Add the broccoli florets to the saucepan, and when the milk comes to a boil, reduce heat, partially cover, and simmer for 20 minutes.
Remove from the heat.
Add the cheese and stir until the cheese melts.